Data 605 Final

CUNY SPS SPRING 2023

Required Libraries

library(data.table)
library(MASS)
library(Matrix)
library(matrixcalc)
library(dplyr)
library(ggplot2)
library(tidyverse)
library(purrr)
library(corrplot)
library(correlation)
library(knitr)
library(Hmisc)
library(forecast)
library(ggplot2)
library(ggthemes)
library(moments)
library(psych)
library(mctest)

Problem 1

Setting up the required Parameters :

# Since we will use the same set of parameters for the 3 PDFs.

#set the seed - using this allows reproducibility of the sequence of random numbers

set.seed(68)


#We are required to choose a value of n > 3

n<- round(runif(1, 4, 100))


#We are required to choose a value of lambda between 2 and 10

lambda <- round(runif(1, 2, 10))


# The number of observations required is given by N :

N <- 10000

cat("We will use random generated values for 'n' and 'lambda' using the 'runif' function ","\n")
## We will use random generated values for 'n' and 'lambda' using the 'runif' function
cat("The random value of n based on the requirement is : ","\n", (n))
## The random value of n based on the requirement is :  
##  93
cat("The random value of lambda based on the requirement is : ", "\n", (lambda))
## The random value of lambda based on the requirement is :  
##  7
cat("The required observations are : ", "\n", (N))
## The required observations are :  
##  10000


Probability Density 1: X~Gamma

Using R, generate a random variable \(X\) that has 10,000 random Gamma Ɣ PDF values. A Gamma Ɣ PDF is completely describe by “n” (a size parameter) and lambda, λ (a shape parameter). Choose any “n” greater than (>) 3 and an expected value (λ) between 2 and 10 (you choose)

#We will use the following function in R: rgamma(n, shape, rate = 1, scale = 1/rate)


cat("For n =",(n), ",  lambda = ",(lambda),"and ",(N),"observations : ")
## For n = 93 ,  lambda =  7 and  10000 observations :
xgamma <- rgamma(N, shape = n, rate = lambda)

cat("The first 10 values of the Gamma PDF are:", "\n", (head(xgamma,10)))
## The first 10 values of the Gamma PDF are: 
##  12.87694 14.14674 12.05287 13.22567 14.84226 13.78659 14.46058 14.3831 13.09145 13.75537


Probability Density 2: Y~Sum of Exponentials

Generate 10,000 observations from the sum of \(n\) exponential PDF with rate/shape parameter (\(\lambda\)). The \(n\) and \(\lambda\) must be the same as in the previous case. (e.g., \(mysum\) \(=\) \(rexp\)(10000,\(\lambda\))+\(rexp\)(10000,\(\lambda\)))

# we will use the following function sum(rexp(n, lambda)), i.e. the sum of the rexp function

cat("For n =",(n), ",  lambda = ",(lambda),"and ",(N),"observations : ")
## For n = 93 ,  lambda =  7 and  10000 observations :
sumexp <- numeric(N)

for (i in 1:N) {
  sumexp[i] <- sum(rexp(n, lambda))
}

cat("The first 10 values of the Sum of Exponentials PDF are : ", "\n", (head(sumexp,10)))
## The first 10 values of the Sum of Exponentials PDF are :  
##  14.79785 12.77006 13.0302 13.45413 12.98259 12.32394 11.55567 13.12414 12.30901 13.78982


Probability Density 3: Z~ Exponential

Generate 10,000 observations from a single exponential pdf with rate/shape parameter (\(\lambda\))

expobs <- rexp(n = N, rate = lambda)

cat("For n =",(n), ",  lambda = ",(lambda),"and ",(N),"observations : ", "\n")
## For n = 93 ,  lambda =  7 and  10000 observations :
cat("The first 10 values of the Exponential PDF are : ", "\n", (head(expobs,10)))
## The first 10 values of the Exponential PDF are :  
##  0.05663102 0.2523144 0.1856502 0.1155885 0.7036552 0.1991529 0.003784598 0.2653726 0.1304988 0.08322686


Problem 1a

Calculate the empirical expected value (means) and variances of all three pdfs

Note : The sample mean and variance are estimates of the population mean and variance, respectively, based on the required sample of 10,000 observations.

# We will use the "mean" and "var" functions in R for this computation

cat("For n =",(n), "and lambda = ",(lambda),"and ",(N),"observations : ", "\n")
## For n = 93 and lambda =  7 and  10000 observations :
cat("The Empirical expected value (mean)  of the Gamma PDF is:", "\n", (mean(xgamma)))
## The Empirical expected value (mean)  of the Gamma PDF is: 
##  13.28273
cat("The Empirical variance of the Gamma PDF is:", "\n", (var(xgamma)))
## The Empirical variance of the Gamma PDF is: 
##  1.913769
cat("\n")
cat("\n")
cat("The Empirical expected value (mean)  of the Sum of Exponentials PDF is:", "\n", (mean(sumexp)))
## The Empirical expected value (mean)  of the Sum of Exponentials PDF is: 
##  13.28817
cat("The Empirical variance of the Sum of Exponentials PDF is:", "\n", (var(sumexp)))
## The Empirical variance of the Sum of Exponentials PDF is: 
##  1.892334
cat("\n")
cat("\n")
cat("The Empirical expected value (mean)  of the Exponential PDF is:", "\n", (mean(expobs)))
## The Empirical expected value (mean)  of the Exponential PDF is: 
##  0.1409875
cat("The Empirical variance of the Sum of the Exponential PDF is:", "\n", (var(expobs)))
## The Empirical variance of the Sum of the Exponential PDF is: 
##  0.02005992
cat("\n")


Problem 1b

Using calculus, calculate the expected value and variance of the Gamma pdf (X). Using the moment generating function for exponentials, calculate the expected value of the single exponential (Z) and the sum of exponentials (Y)

Probability Density 1: X~Gamma

The Gamma Function is defined as :

\(\Gamma(\alpha) = \int_{0}^{\infty}y^{\alpha - 1}e^{-y} dy\)\(for\) \(\alpha \gt 0\)

The Expected Value of the Gamma Function is given as :

\(E(X) = \int_{0}^{\infty} f(x) dx\)

\(\label{eq:gam-mean-s3} \begin{split} \mathrm{E}(X) &= \frac{a}{b} \int_{0}^{\infty} \mathrm{Gam}(x; a+1, b) \, \mathrm{d}x \\&= \frac{a}{b} \; . \end{split}\)

For our Computation, \(a\) = \(n\) = 93, and \(b\) = \(\lambda\) = 7 as was computed above–

\(\implies\) \(E(\Gamma)\) = \(\frac{93}{7}\) = 13.28

Similarly the Variance is given as \(\frac{n}{\lambda^{2}}\) = \(\frac{93}{7^{2}}\) = 1.89

Probability Density 3: Z~ Exponential - Epected Value using MGF

the following proof is from TSingh - Assignment 9

The MGF of the Exponential Distribution is given by ;

\({ g }_{ X }(t)=E({ e }^{ t }X)=\int _{ -\infty }^{ \infty }{ { e }^{ tx }{ f }_{ X }(x)dx. }\)

\(\Rightarrow\) \(g(t)=\frac { λe^{ (t-λ)x } }{ t-λ } |_{ 0 }^{ ∞ }\)

The First Moment

\(\Rightarrow\) \(g'(t)=\frac { λ }{ (λ-t)^{ 2 } }\)

\(\Rightarrow\) \(g'(0)=\frac { λ }{ (λ-0)^{ 2 } }\)

\(\Rightarrow\) \(g'(0) = \frac { λ }{ λ^{ 2 } } =\frac { 1 }{ λ }\)The Expected Value - First Moment

\(\implies\) For our Computation The Expected Value - First Moment =

\(\frac { 1 }{ λ }\) = \(\frac { 1 }{ 7 }\)

= 0.143

Probability Density 2: Y~Sum of Exponentials - Epected Value using MGF

————–#####################————————

1c. Probability

For pdf Z (the exponential), calculate empirically probabilities a through c. Then evaluate through calculus whether the memoryless property holds

a

For \(P(Z>\lambda | Z>\frac{\lambda}{2})\)

Emp_prob_a <- 1-(pexp((mean(expobs)),lambda/2))

Emp_prob_a
## [1] 0.6105126

b

For \(P(Z>2\lambda | Z>\lambda)\)

Emp_prob_b <- 1-(pexp((mean(expobs)),2*lambda))

Emp_prob_b
## [1] 0.1389244

c

For \(P(Z>3\lambda | Z>\lambda)\)

Emp_prob_c <- 1-(pexp((mean(expobs)),3*lambda))

Emp_prob_c
## [1] 0.0517807

1d

Loosely investigate whether P(YZ) = P(Y) P(Z) by building a table with quartiles and evaluating the marginal and joint probabilities


Problem 2

Overview : https://www.kaggle.com/

Compete in the House Prices: Advanced Regression Techniques competition, provide r code for the following requirements :

Descriptive and Inferential Statistics. Provide univariate descriptive statistics and appropriate plots for the training data set. Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. Derive a correlation matrix for any three quantitative variables in the dataset. Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval. Discuss the meaning of your analysis. Would you be worried about familywise error? Why or why not?

Importing the data :

# Import provided datasets stored on Github
p2_train <- data.frame(read.csv('https://raw.githubusercontent.com/tagensingh/sps_data605_final_p_2/main/train.csv', header = T, sep = ","))
p2_test <- data.frame(read.csv('https://raw.githubusercontent.com/tagensingh/sps_data605_final_p_2/main/test.csv', header = T, sep = ","))
p2_test$SalePrice <- 0

###Descriptive Statistics

Provide univariate descriptive statistics and appropriate plots for the training data set

# Summary of dataset : 

kable(head(p2_train))
Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape LandContour Utilities LotConfig LandSlope Neighborhood Condition1 Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold SaleType SaleCondition SalePrice
1 60 RL 65 8450 Pave NA Reg Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2003 2003 Gable CompShg VinylSd VinylSd BrkFace 196 Gd TA PConc Gd TA No GLQ 706 Unf 0 150 856 GasA Ex Y SBrkr 856 854 0 1710 1 0 2 1 3 1 Gd 8 Typ 0 NA Attchd 2003 RFn 2 548 TA TA Y 0 61 0 0 0 0 NA NA NA 0 2 2008 WD Normal 208500
2 20 RL 80 9600 Pave NA Reg Lvl AllPub FR2 Gtl Veenker Feedr Norm 1Fam 1Story 6 8 1976 1976 Gable CompShg MetalSd MetalSd None 0 TA TA CBlock Gd TA Gd ALQ 978 Unf 0 284 1262 GasA Ex Y SBrkr 1262 0 0 1262 0 1 2 0 3 1 TA 6 Typ 1 TA Attchd 1976 RFn 2 460 TA TA Y 298 0 0 0 0 0 NA NA NA 0 5 2007 WD Normal 181500
3 60 RL 68 11250 Pave NA IR1 Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2001 2002 Gable CompShg VinylSd VinylSd BrkFace 162 Gd TA PConc Gd TA Mn GLQ 486 Unf 0 434 920 GasA Ex Y SBrkr 920 866 0 1786 1 0 2 1 3 1 Gd 6 Typ 1 TA Attchd 2001 RFn 2 608 TA TA Y 0 42 0 0 0 0 NA NA NA 0 9 2008 WD Normal 223500
4 70 RL 60 9550 Pave NA IR1 Lvl AllPub Corner Gtl Crawfor Norm Norm 1Fam 2Story 7 5 1915 1970 Gable CompShg Wd Sdng Wd Shng None 0 TA TA BrkTil TA Gd No ALQ 216 Unf 0 540 756 GasA Gd Y SBrkr 961 756 0 1717 1 0 1 0 3 1 Gd 7 Typ 1 Gd Detchd 1998 Unf 3 642 TA TA Y 0 35 272 0 0 0 NA NA NA 0 2 2006 WD Abnorml 140000
5 60 RL 84 14260 Pave NA IR1 Lvl AllPub FR2 Gtl NoRidge Norm Norm 1Fam 2Story 8 5 2000 2000 Gable CompShg VinylSd VinylSd BrkFace 350 Gd TA PConc Gd TA Av GLQ 655 Unf 0 490 1145 GasA Ex Y SBrkr 1145 1053 0 2198 1 0 2 1 4 1 Gd 9 Typ 1 TA Attchd 2000 RFn 3 836 TA TA Y 192 84 0 0 0 0 NA NA NA 0 12 2008 WD Normal 250000
6 50 RL 85 14115 Pave NA IR1 Lvl AllPub Inside Gtl Mitchel Norm Norm 1Fam 1.5Fin 5 5 1993 1995 Gable CompShg VinylSd VinylSd None 0 TA TA Wood Gd TA No GLQ 732 Unf 0 64 796 GasA Ex Y SBrkr 796 566 0 1362 1 0 1 1 1 1 TA 5 Typ 0 NA Attchd 1993 Unf 2 480 TA TA Y 40 30 0 320 0 0 NA MnPrv Shed 700 10 2009 WD Normal 143000
# Dimension of Dataset : Rows X Columns

dim(p2_train)
## [1] 1460   81
# Structure of Columns

str(p2_train)
## 'data.frame':    1460 obs. of  81 variables:
##  $ Id           : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ MSSubClass   : int  60 20 60 70 60 50 20 60 50 190 ...
##  $ MSZoning     : chr  "RL" "RL" "RL" "RL" ...
##  $ LotFrontage  : int  65 80 68 60 84 85 75 NA 51 50 ...
##  $ LotArea      : int  8450 9600 11250 9550 14260 14115 10084 10382 6120 7420 ...
##  $ Street       : chr  "Pave" "Pave" "Pave" "Pave" ...
##  $ Alley        : chr  NA NA NA NA ...
##  $ LotShape     : chr  "Reg" "Reg" "IR1" "IR1" ...
##  $ LandContour  : chr  "Lvl" "Lvl" "Lvl" "Lvl" ...
##  $ Utilities    : chr  "AllPub" "AllPub" "AllPub" "AllPub" ...
##  $ LotConfig    : chr  "Inside" "FR2" "Inside" "Corner" ...
##  $ LandSlope    : chr  "Gtl" "Gtl" "Gtl" "Gtl" ...
##  $ Neighborhood : chr  "CollgCr" "Veenker" "CollgCr" "Crawfor" ...
##  $ Condition1   : chr  "Norm" "Feedr" "Norm" "Norm" ...
##  $ Condition2   : chr  "Norm" "Norm" "Norm" "Norm" ...
##  $ BldgType     : chr  "1Fam" "1Fam" "1Fam" "1Fam" ...
##  $ HouseStyle   : chr  "2Story" "1Story" "2Story" "2Story" ...
##  $ OverallQual  : int  7 6 7 7 8 5 8 7 7 5 ...
##  $ OverallCond  : int  5 8 5 5 5 5 5 6 5 6 ...
##  $ YearBuilt    : int  2003 1976 2001 1915 2000 1993 2004 1973 1931 1939 ...
##  $ YearRemodAdd : int  2003 1976 2002 1970 2000 1995 2005 1973 1950 1950 ...
##  $ RoofStyle    : chr  "Gable" "Gable" "Gable" "Gable" ...
##  $ RoofMatl     : chr  "CompShg" "CompShg" "CompShg" "CompShg" ...
##  $ Exterior1st  : chr  "VinylSd" "MetalSd" "VinylSd" "Wd Sdng" ...
##  $ Exterior2nd  : chr  "VinylSd" "MetalSd" "VinylSd" "Wd Shng" ...
##  $ MasVnrType   : chr  "BrkFace" "None" "BrkFace" "None" ...
##  $ MasVnrArea   : int  196 0 162 0 350 0 186 240 0 0 ...
##  $ ExterQual    : chr  "Gd" "TA" "Gd" "TA" ...
##  $ ExterCond    : chr  "TA" "TA" "TA" "TA" ...
##  $ Foundation   : chr  "PConc" "CBlock" "PConc" "BrkTil" ...
##  $ BsmtQual     : chr  "Gd" "Gd" "Gd" "TA" ...
##  $ BsmtCond     : chr  "TA" "TA" "TA" "Gd" ...
##  $ BsmtExposure : chr  "No" "Gd" "Mn" "No" ...
##  $ BsmtFinType1 : chr  "GLQ" "ALQ" "GLQ" "ALQ" ...
##  $ BsmtFinSF1   : int  706 978 486 216 655 732 1369 859 0 851 ...
##  $ BsmtFinType2 : chr  "Unf" "Unf" "Unf" "Unf" ...
##  $ BsmtFinSF2   : int  0 0 0 0 0 0 0 32 0 0 ...
##  $ BsmtUnfSF    : int  150 284 434 540 490 64 317 216 952 140 ...
##  $ TotalBsmtSF  : int  856 1262 920 756 1145 796 1686 1107 952 991 ...
##  $ Heating      : chr  "GasA" "GasA" "GasA" "GasA" ...
##  $ HeatingQC    : chr  "Ex" "Ex" "Ex" "Gd" ...
##  $ CentralAir   : chr  "Y" "Y" "Y" "Y" ...
##  $ Electrical   : chr  "SBrkr" "SBrkr" "SBrkr" "SBrkr" ...
##  $ X1stFlrSF    : int  856 1262 920 961 1145 796 1694 1107 1022 1077 ...
##  $ X2ndFlrSF    : int  854 0 866 756 1053 566 0 983 752 0 ...
##  $ LowQualFinSF : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ GrLivArea    : int  1710 1262 1786 1717 2198 1362 1694 2090 1774 1077 ...
##  $ BsmtFullBath : int  1 0 1 1 1 1 1 1 0 1 ...
##  $ BsmtHalfBath : int  0 1 0 0 0 0 0 0 0 0 ...
##  $ FullBath     : int  2 2 2 1 2 1 2 2 2 1 ...
##  $ HalfBath     : int  1 0 1 0 1 1 0 1 0 0 ...
##  $ BedroomAbvGr : int  3 3 3 3 4 1 3 3 2 2 ...
##  $ KitchenAbvGr : int  1 1 1 1 1 1 1 1 2 2 ...
##  $ KitchenQual  : chr  "Gd" "TA" "Gd" "Gd" ...
##  $ TotRmsAbvGrd : int  8 6 6 7 9 5 7 7 8 5 ...
##  $ Functional   : chr  "Typ" "Typ" "Typ" "Typ" ...
##  $ Fireplaces   : int  0 1 1 1 1 0 1 2 2 2 ...
##  $ FireplaceQu  : chr  NA "TA" "TA" "Gd" ...
##  $ GarageType   : chr  "Attchd" "Attchd" "Attchd" "Detchd" ...
##  $ GarageYrBlt  : int  2003 1976 2001 1998 2000 1993 2004 1973 1931 1939 ...
##  $ GarageFinish : chr  "RFn" "RFn" "RFn" "Unf" ...
##  $ GarageCars   : int  2 2 2 3 3 2 2 2 2 1 ...
##  $ GarageArea   : int  548 460 608 642 836 480 636 484 468 205 ...
##  $ GarageQual   : chr  "TA" "TA" "TA" "TA" ...
##  $ GarageCond   : chr  "TA" "TA" "TA" "TA" ...
##  $ PavedDrive   : chr  "Y" "Y" "Y" "Y" ...
##  $ WoodDeckSF   : int  0 298 0 0 192 40 255 235 90 0 ...
##  $ OpenPorchSF  : int  61 0 42 35 84 30 57 204 0 4 ...
##  $ EnclosedPorch: int  0 0 0 272 0 0 0 228 205 0 ...
##  $ X3SsnPorch   : int  0 0 0 0 0 320 0 0 0 0 ...
##  $ ScreenPorch  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolArea     : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolQC       : chr  NA NA NA NA ...
##  $ Fence        : chr  NA NA NA NA ...
##  $ MiscFeature  : chr  NA NA NA NA ...
##  $ MiscVal      : int  0 0 0 0 0 700 0 350 0 0 ...
##  $ MoSold       : int  2 5 9 2 12 10 8 11 4 1 ...
##  $ YrSold       : int  2008 2007 2008 2006 2008 2009 2007 2009 2008 2008 ...
##  $ SaleType     : chr  "WD" "WD" "WD" "WD" ...
##  $ SaleCondition: chr  "Normal" "Normal" "Normal" "Abnorml" ...
##  $ SalePrice    : int  208500 181500 223500 140000 250000 143000 307000 200000 129900 118000 ...

###Enhancing the Training Dataset

For our analysis we will convert some categorical columns to numerical values, using the values provided in the description file These Operations will be done on both the Training and Test Datasets

# Duplicating categorical columns and converting to numerical for additional analysis

# Adding Quantified Foundation Column

p2_train$Foundation_q = p2_train$Foundation
p2_train$Foundation_q <- c(Wood=1, Stone=2, Slab=3, PConc=4, CBlock=5, BrkTil=6)[p2_train$Foundation_q]


p2_test$Foundation_q = p2_test$Foundation
p2_test$Foundation_q <- c(Wood=1, Stone=2, Slab=3, PConc=4, CBlock=5, BrkTil=6)[p2_test$Foundation_q]


# Adding Quantified Basement Type Column

p2_train$BsmtFinType2_q = p2_train$BsmtFinType2
p2_train$BsmtFinType2_q <- c(Unf=1, LwQ=2, Rec=3, BLQ=4, ALQ=5, GLQ=6)[p2_train$BsmtFinType2_q]


p2_test$BsmtFinType2_q = p2_test$BsmtFinType2
p2_test$BsmtFinType2_q <- c(Unf=1, LwQ=2, Rec=3, BLQ=4, ALQ=5, GLQ=6)[p2_test$BsmtFinType2_q]


# Adding Quantified Heating Quality Column

p2_train$HeatingQC_q = p2_train$HeatingQC
p2_train$HeatingQC_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_train$HeatingQC_q]   

p2_test$HeatingQC_q = p2_test$HeatingQC
p2_test$HeatingQC_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_test$HeatingQC_q]


# Adding Quantified Electrical Quality Column

p2_train$Electrical_q = p2_train$Electrical
p2_train$Electrical_q <- c(Mix=1, FuseP=2, FuseF=3, FuseA=4, SBrkr=5)[p2_train$Electrical_q]

p2_test$Electrical_q = p2_test$Electrical
p2_test$Electrical_q <- c(Mix=1, FuseP=2, FuseF=3, FuseA=4, SBrkr=5)[p2_test$Electrical_q]


# Adding Quantified KitChen Quality Column

p2_train$KitchenQual_q = p2_train$KitchenQual
p2_train$KitchenQual_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_train$KitchenQual_q]

p2_test$KitchenQual_q = p2_test$KitchenQual
p2_test$KitchenQual_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_test$KitchenQual_q]


# Adding Quantified Garage Condition Column

p2_train$GarageCond_q = p2_train$GarageCond
p2_train$GarageCond_q<- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_train$GarageCond_q]

p2_test$GarageCond_q = p2_test$GarageCond
p2_test$GarageCond_q<- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_test$GarageCond_q]


# Adding Quantified Fence Condition Column

p2_train$Fence_q = p2_train$Fence
p2_train$Fence_q <- c(MnWw=1, GdWo=2, MnPrv=3, GdPrv=4)[p2_train$Fence_q]

p2_test$Fence_q = p2_test$Fence
p2_test$Fence_q <- c(MnWw=1, GdWo=2, MnPrv=3, GdPrv=4)[p2_test$Fence_q]


# Some Cleanup to account for reserved "NA"

p2_train <- replace(p2_train,is.na(p2_train),0)

p2_test <- replace(p2_test,is.na(p2_test),0)


# Printing the enhanced data frame with new quantitative columns

p2_train %>% select(order(colnames(p2_train)))
str(p2_train)
## 'data.frame':    1460 obs. of  88 variables:
##  $ Id            : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ MSSubClass    : int  60 20 60 70 60 50 20 60 50 190 ...
##  $ MSZoning      : chr  "RL" "RL" "RL" "RL" ...
##  $ LotFrontage   : num  65 80 68 60 84 85 75 0 51 50 ...
##  $ LotArea       : int  8450 9600 11250 9550 14260 14115 10084 10382 6120 7420 ...
##  $ Street        : chr  "Pave" "Pave" "Pave" "Pave" ...
##  $ Alley         : chr  "0" "0" "0" "0" ...
##  $ LotShape      : chr  "Reg" "Reg" "IR1" "IR1" ...
##  $ LandContour   : chr  "Lvl" "Lvl" "Lvl" "Lvl" ...
##  $ Utilities     : chr  "AllPub" "AllPub" "AllPub" "AllPub" ...
##  $ LotConfig     : chr  "Inside" "FR2" "Inside" "Corner" ...
##  $ LandSlope     : chr  "Gtl" "Gtl" "Gtl" "Gtl" ...
##  $ Neighborhood  : chr  "CollgCr" "Veenker" "CollgCr" "Crawfor" ...
##  $ Condition1    : chr  "Norm" "Feedr" "Norm" "Norm" ...
##  $ Condition2    : chr  "Norm" "Norm" "Norm" "Norm" ...
##  $ BldgType      : chr  "1Fam" "1Fam" "1Fam" "1Fam" ...
##  $ HouseStyle    : chr  "2Story" "1Story" "2Story" "2Story" ...
##  $ OverallQual   : int  7 6 7 7 8 5 8 7 7 5 ...
##  $ OverallCond   : int  5 8 5 5 5 5 5 6 5 6 ...
##  $ YearBuilt     : int  2003 1976 2001 1915 2000 1993 2004 1973 1931 1939 ...
##  $ YearRemodAdd  : int  2003 1976 2002 1970 2000 1995 2005 1973 1950 1950 ...
##  $ RoofStyle     : chr  "Gable" "Gable" "Gable" "Gable" ...
##  $ RoofMatl      : chr  "CompShg" "CompShg" "CompShg" "CompShg" ...
##  $ Exterior1st   : chr  "VinylSd" "MetalSd" "VinylSd" "Wd Sdng" ...
##  $ Exterior2nd   : chr  "VinylSd" "MetalSd" "VinylSd" "Wd Shng" ...
##  $ MasVnrType    : chr  "BrkFace" "None" "BrkFace" "None" ...
##  $ MasVnrArea    : num  196 0 162 0 350 0 186 240 0 0 ...
##  $ ExterQual     : chr  "Gd" "TA" "Gd" "TA" ...
##  $ ExterCond     : chr  "TA" "TA" "TA" "TA" ...
##  $ Foundation    : chr  "PConc" "CBlock" "PConc" "BrkTil" ...
##  $ BsmtQual      : chr  "Gd" "Gd" "Gd" "TA" ...
##  $ BsmtCond      : chr  "TA" "TA" "TA" "Gd" ...
##  $ BsmtExposure  : chr  "No" "Gd" "Mn" "No" ...
##  $ BsmtFinType1  : chr  "GLQ" "ALQ" "GLQ" "ALQ" ...
##  $ BsmtFinSF1    : int  706 978 486 216 655 732 1369 859 0 851 ...
##  $ BsmtFinType2  : chr  "Unf" "Unf" "Unf" "Unf" ...
##  $ BsmtFinSF2    : int  0 0 0 0 0 0 0 32 0 0 ...
##  $ BsmtUnfSF     : int  150 284 434 540 490 64 317 216 952 140 ...
##  $ TotalBsmtSF   : int  856 1262 920 756 1145 796 1686 1107 952 991 ...
##  $ Heating       : chr  "GasA" "GasA" "GasA" "GasA" ...
##  $ HeatingQC     : chr  "Ex" "Ex" "Ex" "Gd" ...
##  $ CentralAir    : chr  "Y" "Y" "Y" "Y" ...
##  $ Electrical    : chr  "SBrkr" "SBrkr" "SBrkr" "SBrkr" ...
##  $ X1stFlrSF     : int  856 1262 920 961 1145 796 1694 1107 1022 1077 ...
##  $ X2ndFlrSF     : int  854 0 866 756 1053 566 0 983 752 0 ...
##  $ LowQualFinSF  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ GrLivArea     : int  1710 1262 1786 1717 2198 1362 1694 2090 1774 1077 ...
##  $ BsmtFullBath  : int  1 0 1 1 1 1 1 1 0 1 ...
##  $ BsmtHalfBath  : int  0 1 0 0 0 0 0 0 0 0 ...
##  $ FullBath      : int  2 2 2 1 2 1 2 2 2 1 ...
##  $ HalfBath      : int  1 0 1 0 1 1 0 1 0 0 ...
##  $ BedroomAbvGr  : int  3 3 3 3 4 1 3 3 2 2 ...
##  $ KitchenAbvGr  : int  1 1 1 1 1 1 1 1 2 2 ...
##  $ KitchenQual   : chr  "Gd" "TA" "Gd" "Gd" ...
##  $ TotRmsAbvGrd  : int  8 6 6 7 9 5 7 7 8 5 ...
##  $ Functional    : chr  "Typ" "Typ" "Typ" "Typ" ...
##  $ Fireplaces    : int  0 1 1 1 1 0 1 2 2 2 ...
##  $ FireplaceQu   : chr  "0" "TA" "TA" "Gd" ...
##  $ GarageType    : chr  "Attchd" "Attchd" "Attchd" "Detchd" ...
##  $ GarageYrBlt   : num  2003 1976 2001 1998 2000 ...
##  $ GarageFinish  : chr  "RFn" "RFn" "RFn" "Unf" ...
##  $ GarageCars    : int  2 2 2 3 3 2 2 2 2 1 ...
##  $ GarageArea    : int  548 460 608 642 836 480 636 484 468 205 ...
##  $ GarageQual    : chr  "TA" "TA" "TA" "TA" ...
##  $ GarageCond    : chr  "TA" "TA" "TA" "TA" ...
##  $ PavedDrive    : chr  "Y" "Y" "Y" "Y" ...
##  $ WoodDeckSF    : int  0 298 0 0 192 40 255 235 90 0 ...
##  $ OpenPorchSF   : int  61 0 42 35 84 30 57 204 0 4 ...
##  $ EnclosedPorch : int  0 0 0 272 0 0 0 228 205 0 ...
##  $ X3SsnPorch    : int  0 0 0 0 0 320 0 0 0 0 ...
##  $ ScreenPorch   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolArea      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolQC        : chr  "0" "0" "0" "0" ...
##  $ Fence         : chr  "0" "0" "0" "0" ...
##  $ MiscFeature   : chr  "0" "0" "0" "0" ...
##  $ MiscVal       : int  0 0 0 0 0 700 0 350 0 0 ...
##  $ MoSold        : int  2 5 9 2 12 10 8 11 4 1 ...
##  $ YrSold        : int  2008 2007 2008 2006 2008 2009 2007 2009 2008 2008 ...
##  $ SaleType      : chr  "WD" "WD" "WD" "WD" ...
##  $ SaleCondition : chr  "Normal" "Normal" "Normal" "Abnorml" ...
##  $ SalePrice     : int  208500 181500 223500 140000 250000 143000 307000 200000 129900 118000 ...
##  $ Foundation_q  : num  4 5 4 6 4 1 4 5 6 6 ...
##  $ BsmtFinType2_q: num  1 1 1 1 1 1 1 4 1 1 ...
##  $ HeatingQC_q   : num  5 5 5 4 5 5 5 5 4 5 ...
##  $ Electrical_q  : num  5 5 5 5 5 5 5 5 3 5 ...
##  $ KitchenQual_q : num  4 3 4 4 4 3 4 3 3 3 ...
##  $ GarageCond_q  : num  3 3 3 3 3 3 3 3 3 3 ...
##  $ Fence_q       : num  0 0 0 0 0 3 0 0 0 0 ...
kable(head(p2_train))
Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape LandContour Utilities LotConfig LandSlope Neighborhood Condition1 Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold SaleType SaleCondition SalePrice Foundation_q BsmtFinType2_q HeatingQC_q Electrical_q KitchenQual_q GarageCond_q Fence_q
1 60 RL 65 8450 Pave 0 Reg Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2003 2003 Gable CompShg VinylSd VinylSd BrkFace 196 Gd TA PConc Gd TA No GLQ 706 Unf 0 150 856 GasA Ex Y SBrkr 856 854 0 1710 1 0 2 1 3 1 Gd 8 Typ 0 0 Attchd 2003 RFn 2 548 TA TA Y 0 61 0 0 0 0 0 0 0 0 2 2008 WD Normal 208500 4 1 5 5 4 3 0
2 20 RL 80 9600 Pave 0 Reg Lvl AllPub FR2 Gtl Veenker Feedr Norm 1Fam 1Story 6 8 1976 1976 Gable CompShg MetalSd MetalSd None 0 TA TA CBlock Gd TA Gd ALQ 978 Unf 0 284 1262 GasA Ex Y SBrkr 1262 0 0 1262 0 1 2 0 3 1 TA 6 Typ 1 TA Attchd 1976 RFn 2 460 TA TA Y 298 0 0 0 0 0 0 0 0 0 5 2007 WD Normal 181500 5 1 5 5 3 3 0
3 60 RL 68 11250 Pave 0 IR1 Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2001 2002 Gable CompShg VinylSd VinylSd BrkFace 162 Gd TA PConc Gd TA Mn GLQ 486 Unf 0 434 920 GasA Ex Y SBrkr 920 866 0 1786 1 0 2 1 3 1 Gd 6 Typ 1 TA Attchd 2001 RFn 2 608 TA TA Y 0 42 0 0 0 0 0 0 0 0 9 2008 WD Normal 223500 4 1 5 5 4 3 0
4 70 RL 60 9550 Pave 0 IR1 Lvl AllPub Corner Gtl Crawfor Norm Norm 1Fam 2Story 7 5 1915 1970 Gable CompShg Wd Sdng Wd Shng None 0 TA TA BrkTil TA Gd No ALQ 216 Unf 0 540 756 GasA Gd Y SBrkr 961 756 0 1717 1 0 1 0 3 1 Gd 7 Typ 1 Gd Detchd 1998 Unf 3 642 TA TA Y 0 35 272 0 0 0 0 0 0 0 2 2006 WD Abnorml 140000 6 1 4 5 4 3 0
5 60 RL 84 14260 Pave 0 IR1 Lvl AllPub FR2 Gtl NoRidge Norm Norm 1Fam 2Story 8 5 2000 2000 Gable CompShg VinylSd VinylSd BrkFace 350 Gd TA PConc Gd TA Av GLQ 655 Unf 0 490 1145 GasA Ex Y SBrkr 1145 1053 0 2198 1 0 2 1 4 1 Gd 9 Typ 1 TA Attchd 2000 RFn 3 836 TA TA Y 192 84 0 0 0 0 0 0 0 0 12 2008 WD Normal 250000 4 1 5 5 4 3 0
6 50 RL 85 14115 Pave 0 IR1 Lvl AllPub Inside Gtl Mitchel Norm Norm 1Fam 1.5Fin 5 5 1993 1995 Gable CompShg VinylSd VinylSd None 0 TA TA Wood Gd TA No GLQ 732 Unf 0 64 796 GasA Ex Y SBrkr 796 566 0 1362 1 0 1 1 1 1 TA 5 Typ 0 0 Attchd 1993 Unf 2 480 TA TA Y 40 30 0 320 0 0 0 MnPrv Shed 700 10 2009 WD Normal 143000 1 1 5 5 3 3 3

Investigative Scatter Plots

p2_train$OverallCond_factor <- as.factor(as.character(p2_train$OverallCond))
ggplot(p2_train, aes(x=OverallCond, y=SalePrice, fill=OverallCond_factor)) + geom_boxplot()

p2_train$OverallCond_factor<-NULL
ggplot(p2_train, aes(x=Neighborhood, y=SalePrice, fill=Neighborhood)) + geom_boxplot()+ coord_flip()

A graphical view of the Sales Price Spread vs Year Remodeled Note that We used the year Remodeled vs Year Built since for the homes that were not Remodeled, the year built was used.

ggplot(p2_train, aes(x = YearRemodAdd, y = SalePrice)) +
  geom_point()+
  geom_smooth(method=lm) +
  scale_y_continuous(labels = scales::comma)
## `geom_smooth()` using formula = 'y ~ x'

Correlation Matricies

Derive a correlation matrix for any three quantitative variables in the dataset

Note: as a step further we will compute the Correlation Matrix for a range of quantitative variables below

corr_data<-dplyr::select(p2_train,SalePrice,LotArea,BsmtFinSF2,GarageArea,YearRemodAdd,OverallCond,TotalBsmtSF,GrLivArea,HeatingQC_q,Electrical_q,KitchenQual_q,Fence_q,GarageCond_q)

corr_matrix<-round(cor(corr_data),4)

#Correlation Matrix with correlation matrix coefficients
corrplot(corr_matrix, method = 'number') # colorful number

# Another Visual of the Correlation Matrix

corrplot(corr_matrix, order = 'hclust', addrect = 2)


Correlation Hypothesis Testing

We are required to compute 3 pairs, we will compute an additional 3 pairs to solidify the concept

Sales Price Vs Year Remodeled

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.481 and 0.531. The sample estimate is +0.51

cor.test(corr_data$SalePrice,corr_data$YearRemodAdd, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_data$SalePrice and corr_data$YearRemodAdd
## t = 22.466, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.4817381 0.5316150
## sample estimates:
##      cor 
## 0.507101

Sales Price Vs Lot Area

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.232 and 0.294 The sample estimate is +0.26

cor.test(corr_data$SalePrice,corr_data$LotArea, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_data$SalePrice and corr_data$LotArea
## t = 10.445, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.2323391 0.2947946
## sample estimates:
##       cor 
## 0.2638434

Sales Price Vs Overall Condition

This pair of variable computes a low P-Value (0.002) indication a likely non zero(0) correlation and 80% confidence that the correlation is between -0.111 and -0.044 The sample estimate is -0.07, This indicates a zero to slight inverse relationship between Sales Price and Overall Condition.

cor.test(corr_data$SalePrice,corr_data$OverallCond, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_data$SalePrice and corr_data$OverallCond
## t = -2.9819, df = 1458, p-value = 0.002912
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  -0.1111272 -0.0444103
## sample estimates:
##         cor 
## -0.07785589

Sales Price Vs Quality of Heating System

The Heating Quality variable is derived from converting a categorical field to a numeric

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.399 and 0.454 The sample estimate is +0.427

cor.test(corr_data$SalePrice,corr_data$HeatingQC_q, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_data$SalePrice and corr_data$HeatingQC_q
## t = 18.064, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.3998256 0.4546844
## sample estimates:
##       cor 
## 0.4276487

Sales Price Vs Kitchen Quality

The Kitchen Quality variable is derived from converting a categorical field to a numeric

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.640 and 0.678 The sample estimate is +0.659

cor.test(corr_data$SalePrice,corr_data$KitchenQual_q, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_data$SalePrice and corr_data$KitchenQual_q
## t = 33.509, df = 1458, p-value < 2.2e-16
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  0.6402106 0.6781490
## sample estimates:
##       cor 
## 0.6595997

Sales Price Vs Fence Condition

The Fence Quality variable is derived from converting a categorical field to a numeric

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between -0.179 and -0.113 The sample estimate is -0.146, This indicates an inverse relationship between Sales Price and Overall Condition

cor.test(corr_data$SalePrice,corr_data$Fence_q, conf.level = 0.8)
## 
##  Pearson's product-moment correlation
## 
## data:  corr_data$SalePrice and corr_data$Fence_q
## t = -5.6724, df = 1458, p-value = 1.696e-08
## alternative hypothesis: true correlation is not equal to 0
## 80 percent confidence interval:
##  -0.1796174 -0.1139418
## sample estimates:
##        cor 
## -0.1469415

Pairwise Correlation Discussion

The six correlation analysis pairs of variables show that correlation exist between the Sales Price (Dependent Variable) and the Independent Variables examined. There are some strong correlation in the 80% confidence interval except for the “Fence Condition” and “Overall Condition” Variables which indicated a zero to slight inverse relationship to the Sales Price. This quantitative evidence is not worrying with respect to familywise errors.


Linear Algebra and Correlation

Inverting the Correlation Matrix to Create Precision Matrix

##    The Current Correlation Matrix

corr_matrix
##               SalePrice LotArea BsmtFinSF2 GarageArea YearRemodAdd OverallCond
## SalePrice        1.0000  0.2638    -0.0114     0.6234       0.5071     -0.0779
## LotArea          0.2638  1.0000     0.1112     0.1804       0.0138     -0.0056
## BsmtFinSF2      -0.0114  0.1112     1.0000    -0.0182      -0.0678      0.0402
## GarageArea       0.6234  0.1804    -0.0182     1.0000       0.3716     -0.1515
## YearRemodAdd     0.5071  0.0138    -0.0678     0.3716       1.0000      0.0737
## OverallCond     -0.0779 -0.0056     0.0402    -0.1515       0.0737      1.0000
## TotalBsmtSF      0.6136  0.2608     0.1048     0.4867       0.2911     -0.1711
## GrLivArea        0.7086  0.2631    -0.0096     0.4690       0.2874     -0.0797
## HeatingQC_q      0.4276  0.0036    -0.0745     0.2955       0.5500     -0.0141
## Electrical_q     0.2236  0.0458     0.0292     0.2141       0.3141      0.0973
## KitchenQual_q    0.6596  0.0679    -0.0451     0.4896       0.6253     -0.0267
## Fence_q         -0.1469 -0.0414     0.1153    -0.1228      -0.1411      0.1697
## GarageCond_q     0.2632  0.0761     0.0444     0.5473       0.1441      0.0167
##               TotalBsmtSF GrLivArea HeatingQC_q Electrical_q KitchenQual_q
## SalePrice          0.6136    0.7086      0.4276       0.2236        0.6596
## LotArea            0.2608    0.2631      0.0036       0.0458        0.0679
## BsmtFinSF2         0.1048   -0.0096     -0.0745       0.0292       -0.0451
## GarageArea         0.4867    0.4690      0.2955       0.2141        0.4896
## YearRemodAdd       0.2911    0.2874      0.5500       0.3141        0.6253
## OverallCond       -0.1711   -0.0797     -0.0141       0.0973       -0.0267
## TotalBsmtSF        1.0000    0.4549      0.2657       0.1803        0.4326
## GrLivArea          0.4549    1.0000      0.2546       0.1211        0.4206
## HeatingQC_q        0.2657    0.2546      1.0000       0.1861        0.5042
## Electrical_q       0.1803    0.1211      0.1861       1.0000        0.2313
## KitchenQual_q      0.4326    0.4206      0.5042       0.2313        1.0000
## Fence_q           -0.1094   -0.0784     -0.1803       0.0218       -0.1317
## GarageCond_q       0.1766    0.1533      0.1553       0.1889        0.2321
##               Fence_q GarageCond_q
## SalePrice     -0.1469       0.2632
## LotArea       -0.0414       0.0761
## BsmtFinSF2     0.1153       0.0444
## GarageArea    -0.1228       0.5473
## YearRemodAdd  -0.1411       0.1441
## OverallCond    0.1697       0.0167
## TotalBsmtSF   -0.1094       0.1766
## GrLivArea     -0.0784       0.1533
## HeatingQC_q   -0.1803       0.1553
## Electrical_q   0.0218       0.1889
## KitchenQual_q -0.1317       0.2321
## Fence_q        1.0000       0.0030
## GarageCond_q   0.0030       1.0000
##    Inverting the correlation Matrix to Create the precision Matrix

Invert_matrix<-round(solve(corr_matrix),4)


##   Creating the Precision Matrix 

###   corr X invert

precision_matrix_1 <- round(corr_matrix %*% Invert_matrix,4)

precision_matrix_1
##               SalePrice LotArea BsmtFinSF2 GarageArea YearRemodAdd OverallCond
## SalePrice        0.9999  -1e-04          0      0e+00            0      -1e-04
## LotArea         -0.0001   1e+00          0      0e+00            0      -1e-04
## BsmtFinSF2       0.0000   0e+00          1      0e+00            0       0e+00
## GarageArea      -0.0001  -1e-04          0      1e+00            0      -1e-04
## YearRemodAdd    -0.0001   0e+00          0      0e+00            1      -1e-04
## OverallCond      0.0000   0e+00          0      0e+00            0       1e+00
## TotalBsmtSF     -0.0001  -1e-04          0      0e+00            0      -1e-04
## GrLivArea       -0.0001   0e+00          0      0e+00            0       0e+00
## HeatingQC_q      0.0000   0e+00          0      0e+00            0       0e+00
## Electrical_q     0.0000   0e+00          0      0e+00            0       0e+00
## KitchenQual_q   -0.0001  -1e-04          0      0e+00            0      -1e-04
## Fence_q          0.0000   0e+00          0      0e+00            0       0e+00
## GarageCond_q    -0.0001  -1e-04          0     -1e-04            0       0e+00
##               TotalBsmtSF GrLivArea HeatingQC_q Electrical_q KitchenQual_q
## SalePrice               0         0           0            0             0
## LotArea                 0         0           0            0             0
## BsmtFinSF2              0         0           0            0             0
## GarageArea              0         0           0            0             0
## YearRemodAdd            0         0           0            0             0
## OverallCond             0         0           0            0             0
## TotalBsmtSF             1         0           0            0             0
## GrLivArea               0         1           0            0             0
## HeatingQC_q             0         0           1            0             0
## Electrical_q            0         0           0            1             0
## KitchenQual_q           0         0           0            0             1
## Fence_q                 0         0           0            0             0
## GarageCond_q            0         0           0            0             0
##               Fence_q GarageCond_q
## SalePrice       0e+00       -1e-04
## LotArea         0e+00       -1e-04
## BsmtFinSF2      0e+00        0e+00
## GarageArea      0e+00       -1e-04
## YearRemodAdd    0e+00       -1e-04
## OverallCond     0e+00        0e+00
## TotalBsmtSF     0e+00       -1e-04
## GrLivArea       0e+00       -1e-04
## HeatingQC_q     0e+00       -1e-04
## Electrical_q    0e+00       -1e-04
## KitchenQual_q  -1e-04       -1e-04
## Fence_q         1e+00        0e+00
## GarageCond_q    0e+00        1e+00
###   invert X corr

precision_matrix_2 <- round(Invert_matrix %*% corr_matrix,4)

precision_matrix_2
##               SalePrice LotArea BsmtFinSF2 GarageArea YearRemodAdd OverallCond
## SalePrice        0.9999  -1e-04          0     -1e-04       -1e-04           0
## LotArea         -0.0001   1e+00          0     -1e-04        0e+00           0
## BsmtFinSF2       0.0000   0e+00          1      0e+00        0e+00           0
## GarageArea       0.0000   0e+00          0      1e+00        0e+00           0
## YearRemodAdd     0.0000   0e+00          0      0e+00        1e+00           0
## OverallCond     -0.0001  -1e-04          0     -1e-04       -1e-04           1
## TotalBsmtSF      0.0000   0e+00          0      0e+00        0e+00           0
## GrLivArea        0.0000   0e+00          0      0e+00        0e+00           0
## HeatingQC_q      0.0000   0e+00          0      0e+00        0e+00           0
## Electrical_q     0.0000   0e+00          0      0e+00        0e+00           0
## KitchenQual_q    0.0000   0e+00          0      0e+00        0e+00           0
## Fence_q          0.0000   0e+00          0      0e+00        0e+00           0
## GarageCond_q    -0.0001  -1e-04          0     -1e-04       -1e-04           0
##               TotalBsmtSF GrLivArea HeatingQC_q Electrical_q KitchenQual_q
## SalePrice          -1e-04    -1e-04       0e+00        0e+00        -1e-04
## LotArea            -1e-04     0e+00       0e+00        0e+00        -1e-04
## BsmtFinSF2          0e+00     0e+00       0e+00        0e+00         0e+00
## GarageArea          0e+00     0e+00       0e+00        0e+00         0e+00
## YearRemodAdd        0e+00     0e+00       0e+00        0e+00         0e+00
## OverallCond        -1e-04     0e+00       0e+00        0e+00        -1e-04
## TotalBsmtSF         1e+00     0e+00       0e+00        0e+00         0e+00
## GrLivArea           0e+00     1e+00       0e+00        0e+00         0e+00
## HeatingQC_q         0e+00     0e+00       1e+00        0e+00         0e+00
## Electrical_q        0e+00     0e+00       0e+00        1e+00         0e+00
## KitchenQual_q       0e+00     0e+00       0e+00        0e+00         1e+00
## Fence_q             0e+00     0e+00       0e+00        0e+00        -1e-04
## GarageCond_q       -1e-04    -1e-04      -1e-04       -1e-04        -1e-04
##               Fence_q GarageCond_q
## SalePrice           0       -1e-04
## LotArea             0       -1e-04
## BsmtFinSF2          0        0e+00
## GarageArea          0       -1e-04
## YearRemodAdd        0        0e+00
## OverallCond         0        0e+00
## TotalBsmtSF         0        0e+00
## GrLivArea           0        0e+00
## HeatingQC_q         0        0e+00
## Electrical_q        0        0e+00
## KitchenQual_q       0        0e+00
## Fence_q             1        0e+00
## GarageCond_q        0        1e+00
###   The Decomposition of precision_matrix_1 is :

decomp_pm1 <- lu.decomposition(precision_matrix_1)

decomp_pm1
## $L
##              [,1]        [,2] [,3]   [,4] [,5]        [,6] [,7] [,8] [,9] [,10]
##  [1,]  1.00000000  0.0000e+00    0  0e+00    0  0.0000e+00    0    0    0     0
##  [2,] -0.00010001  1.0000e+00    0  0e+00    0  0.0000e+00    0    0    0     0
##  [3,]  0.00000000  0.0000e+00    1  0e+00    0  0.0000e+00    0    0    0     0
##  [4,] -0.00010001 -1.0001e-04    0  1e+00    0  0.0000e+00    0    0    0     0
##  [5,] -0.00010001 -1.0001e-08    0  0e+00    1  0.0000e+00    0    0    0     0
##  [6,]  0.00000000  0.0000e+00    0  0e+00    0  1.0000e+00    0    0    0     0
##  [7,] -0.00010001 -1.0001e-04    0  0e+00    0 -1.0002e-04    1    0    0     0
##  [8,] -0.00010001 -1.0001e-08    0  0e+00    0 -1.0002e-08    0    1    0     0
##  [9,]  0.00000000  0.0000e+00    0  0e+00    0  0.0000e+00    0    0    1     0
## [10,]  0.00000000  0.0000e+00    0  0e+00    0  0.0000e+00    0    0    0     1
## [11,] -0.00010001 -1.0001e-04    0  0e+00    0 -1.0002e-04    0    0    0     0
## [12,]  0.00000000  0.0000e+00    0  0e+00    0  0.0000e+00    0    0    0     0
## [13,] -0.00010001 -1.0001e-04    0 -1e-04    0 -3.0005e-08    0    0    0     0
##       [,11] [,12] [,13]
##  [1,]     0     0     0
##  [2,]     0     0     0
##  [3,]     0     0     0
##  [4,]     0     0     0
##  [5,]     0     0     0
##  [6,]     0     0     0
##  [7,]     0     0     0
##  [8,]     0     0     0
##  [9,]     0     0     0
## [10,]     0     0     0
## [11,]     1     0     0
## [12,]     0     1     0
## [13,]     0     0     1
## 
## $U
##         [,1]   [,2] [,3] [,4] [,5]        [,6] [,7] [,8] [,9] [,10] [,11]
##  [1,] 0.9999 -1e-04    0    0    0 -0.00010000    0    0    0     0     0
##  [2,] 0.0000  1e+00    0    0    0 -0.00010001    0    0    0     0     0
##  [3,] 0.0000  0e+00    1    0    0  0.00000000    0    0    0     0     0
##  [4,] 0.0000  0e+00    0    1    0 -0.00010002    0    0    0     0     0
##  [5,] 0.0000  0e+00    0    0    1 -0.00010001    0    0    0     0     0
##  [6,] 0.0000  0e+00    0    0    0  1.00000000    0    0    0     0     0
##  [7,] 0.0000  0e+00    0    0    0  0.00000000    1    0    0     0     0
##  [8,] 0.0000  0e+00    0    0    0  0.00000000    0    1    0     0     0
##  [9,] 0.0000  0e+00    0    0    0  0.00000000    0    0    1     0     0
## [10,] 0.0000  0e+00    0    0    0  0.00000000    0    0    0     1     0
## [11,] 0.0000  0e+00    0    0    0  0.00000000    0    0    0     0     1
## [12,] 0.0000  0e+00    0    0    0  0.00000000    0    0    0     0     0
## [13,] 0.0000  0e+00    0    0    0  0.00000000    0    0    0     0     0
##        [,12]       [,13]
##  [1,]  0e+00 -0.00010000
##  [2,]  0e+00 -0.00010001
##  [3,]  0e+00  0.00000000
##  [4,]  0e+00 -0.00010002
##  [5,]  0e+00 -0.00010001
##  [6,]  0e+00  0.00000000
##  [7,]  0e+00 -0.00010002
##  [8,]  0e+00 -0.00010001
##  [9,]  0e+00 -0.00010000
## [10,]  0e+00 -0.00010000
## [11,] -1e-04 -0.00010002
## [12,]  1e+00  0.00000000
## [13,]  0e+00  0.99999997
###   The Decomposition of precision_matrix_1 is :

decomp_pm2 <- lu.decomposition(precision_matrix_2)

decomp_pm2
## $L
##              [,1]        [,2] [,3]        [,4]        [,5] [,6]        [,7]
##  [1,]  1.00000000  0.00000000    0  0.00000000  0.00000000    0  0.00000000
##  [2,] -0.00010001  1.00000000    0  0.00000000  0.00000000    0  0.00000000
##  [3,]  0.00000000  0.00000000    1  0.00000000  0.00000000    0  0.00000000
##  [4,]  0.00000000  0.00000000    0  1.00000000  0.00000000    0  0.00000000
##  [5,]  0.00000000  0.00000000    0  0.00000000  1.00000000    0  0.00000000
##  [6,] -0.00010001 -0.00010001    0 -0.00010002 -0.00010001    1  0.00000000
##  [7,]  0.00000000  0.00000000    0  0.00000000  0.00000000    0  1.00000000
##  [8,]  0.00000000  0.00000000    0  0.00000000  0.00000000    0  0.00000000
##  [9,]  0.00000000  0.00000000    0  0.00000000  0.00000000    0  0.00000000
## [10,]  0.00000000  0.00000000    0  0.00000000  0.00000000    0  0.00000000
## [11,]  0.00000000  0.00000000    0  0.00000000  0.00000000    0  0.00000000
## [12,]  0.00000000  0.00000000    0  0.00000000  0.00000000    0  0.00000000
## [13,] -0.00010001 -0.00010001    0 -0.00010002 -0.00010001    0 -0.00010002
##              [,8]   [,9]  [,10]       [,11] [,12] [,13]
##  [1,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [2,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [3,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [4,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [5,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [6,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [7,]  0.00000000  0e+00  0e+00  0.00000000     0     0
##  [8,]  1.00000000  0e+00  0e+00  0.00000000     0     0
##  [9,]  0.00000000  1e+00  0e+00  0.00000000     0     0
## [10,]  0.00000000  0e+00  1e+00  0.00000000     0     0
## [11,]  0.00000000  0e+00  0e+00  1.00000000     0     0
## [12,]  0.00000000  0e+00  0e+00 -0.00010000     1     0
## [13,] -0.00010001 -1e-04 -1e-04 -0.00010002     0     1
## 
## $U
##         [,1]   [,2] [,3]        [,4]        [,5] [,6]        [,7]        [,8]
##  [1,] 0.9999 -1e-04    0 -0.00010000 -1.0000e-04    0 -0.00010000 -1.0000e-04
##  [2,] 0.0000  1e+00    0 -0.00010001 -1.0001e-08    0 -0.00010001 -1.0001e-08
##  [3,] 0.0000  0e+00    1  0.00000000  0.0000e+00    0  0.00000000  0.0000e+00
##  [4,] 0.0000  0e+00    0  1.00000000  0.0000e+00    0  0.00000000  0.0000e+00
##  [5,] 0.0000  0e+00    0  0.00000000  1.0000e+00    0  0.00000000  0.0000e+00
##  [6,] 0.0000  0e+00    0  0.00000000  0.0000e+00    1 -0.00010002 -1.0002e-08
##  [7,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  1.00000000  0.0000e+00
##  [8,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  0.00000000  1.0000e+00
##  [9,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  0.00000000  0.0000e+00
## [10,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  0.00000000  0.0000e+00
## [11,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  0.00000000  0.0000e+00
## [12,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  0.00000000  0.0000e+00
## [13,] 0.0000  0e+00    0  0.00000000  0.0000e+00    0  0.00000000  0.0000e+00
##       [,9] [,10]       [,11] [,12]       [,13]
##  [1,]    0     0 -0.00010000     0 -1.0000e-04
##  [2,]    0     0 -0.00010001     0 -1.0001e-04
##  [3,]    0     0  0.00000000     0  0.0000e+00
##  [4,]    0     0  0.00000000     0 -1.0000e-04
##  [5,]    0     0  0.00000000     0  0.0000e+00
##  [6,]    0     0 -0.00010002     0 -3.0005e-08
##  [7,]    0     0  0.00000000     0  0.0000e+00
##  [8,]    0     0  0.00000000     0  0.0000e+00
##  [9,]    1     0  0.00000000     0  0.0000e+00
## [10,]    0     1  0.00000000     0  0.0000e+00
## [11,]    0     0  1.00000000     0  0.0000e+00
## [12,]    0     0  0.00000000     1  0.0000e+00
## [13,]    0     0  0.00000000     0  1.0000e+00
## Comparing the 2 Matrices, we see that they are equal ( when rounded to 3 decimal places)

round(precision_matrix_1 ,3)== round(precision_matrix_2,3)
##               SalePrice LotArea BsmtFinSF2 GarageArea YearRemodAdd OverallCond
## SalePrice          TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## LotArea            TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## BsmtFinSF2         TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## GarageArea         TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## YearRemodAdd       TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## OverallCond        TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## TotalBsmtSF        TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## GrLivArea          TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## HeatingQC_q        TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## Electrical_q       TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## KitchenQual_q      TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## Fence_q            TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
## GarageCond_q       TRUE    TRUE       TRUE       TRUE         TRUE        TRUE
##               TotalBsmtSF GrLivArea HeatingQC_q Electrical_q KitchenQual_q
## SalePrice            TRUE      TRUE        TRUE         TRUE          TRUE
## LotArea              TRUE      TRUE        TRUE         TRUE          TRUE
## BsmtFinSF2           TRUE      TRUE        TRUE         TRUE          TRUE
## GarageArea           TRUE      TRUE        TRUE         TRUE          TRUE
## YearRemodAdd         TRUE      TRUE        TRUE         TRUE          TRUE
## OverallCond          TRUE      TRUE        TRUE         TRUE          TRUE
## TotalBsmtSF          TRUE      TRUE        TRUE         TRUE          TRUE
## GrLivArea            TRUE      TRUE        TRUE         TRUE          TRUE
## HeatingQC_q          TRUE      TRUE        TRUE         TRUE          TRUE
## Electrical_q         TRUE      TRUE        TRUE         TRUE          TRUE
## KitchenQual_q        TRUE      TRUE        TRUE         TRUE          TRUE
## Fence_q              TRUE      TRUE        TRUE         TRUE          TRUE
## GarageCond_q         TRUE      TRUE        TRUE         TRUE          TRUE
##               Fence_q GarageCond_q
## SalePrice        TRUE         TRUE
## LotArea          TRUE         TRUE
## BsmtFinSF2       TRUE         TRUE
## GarageArea       TRUE         TRUE
## YearRemodAdd     TRUE         TRUE
## OverallCond      TRUE         TRUE
## TotalBsmtSF      TRUE         TRUE
## GrLivArea        TRUE         TRUE
## HeatingQC_q      TRUE         TRUE
## Electrical_q     TRUE         TRUE
## KitchenQual_q    TRUE         TRUE
## Fence_q          TRUE         TRUE
## GarageCond_q     TRUE         TRUE

Calculus-Based Probability & Statistics

Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary

head(corr_data)
skew(corr_data, na.rm = TRUE)
##  [1]  1.8790086 12.1826150  4.2465214  0.1796113 -0.5025278  0.6916440
##  [7]  1.5211239  1.3637536 -0.5393477 -4.7209627  0.3859710  1.8034393
## [13] -3.3250565
## We see that "LotArea" field is the most RIGHT skewed with a value of :

round(skew(corr_data$LotArea, na.rm = TRUE),3)
## [1] 12.183
## A Histogram of the field :


ggplot(corr_data, aes(x=LotArea)) + geom_histogram(color="blue", fill="white", binwidth = 1000)+labs(title="Lot Area plot - Skewness = 12.2 - Min Value = 1300",x="Lot Size", y = "Count")

The Fitting

Then load the MASS package and run fitdistr to fit an exponential probability density function

la_fit <- corr_data$LotArea

summary(la_fit) ### Note that the minimum value is > 0
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    1300    7554    9478   10517   11602  215245
###  Determining the fit

fit <- fitdistr(la_fit, "exponential")

fit
##        rate    
##   9.508570e-05 
##  (2.488507e-06)

Find the optimal value of λ for this distribution, and then take 1000 samples from this exponential distribution using this value

# Computing Lambda
lambda_fit <- fit$estimate

lambda_fit
##        rate 
## 9.50857e-05
### Generating new distribution and Histogram 

new_dist <- rexp(1000, lambda_fit)

summary(new_dist)
##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
##     8.85  2935.45  7905.62 10876.57 15249.17 81694.21
hist(new_dist,breaks = 100)

Plot histogram and compare it with original histogram

Note: As shown below, using the Lambda from the “LotArea” variable and applying it a new Exponential distribution yields a similar histogram, the differences in distribution values has some effect on the resulting histogram but not significant.

fit_df <- data.frame(length = la_fit)
new_dist_df <- data.frame(length = new_dist)

fit_df$from <- 'Fit'
new_dist_df$from <- 'New Dist'

both_df <- rbind(fit_df,new_dist_df)

ggplot(both_df, aes(length, fill = from)) + geom_density(alpha = 0.5)


Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.

The Exponential PDF is given as \(f(x;\lambda) = \lambda e^{-\lambda x}\) for \(x \geq 0\)

The CDF is given as \(f(x;\lambda)=1−e^{-\lambda x}\)

\(\lambda\) is given as : 9.50857

To find the \(5^{th}\) percentile we solve for x in :

\(0.05 = 1 - e^{\lambda x}\)

\(\implies\) \(0.05 = 1 - e^{-\lambda x}\)

\(\implies\) \(-ln(0.95) = \lambda x\)

\(\implies\) $ x = $

To find the \(95^{th}\) percentile we solve for x in :

\(0.95 = 1 - e^{\lambda x}\)

\(\implies\) \(0.95 = 1 - e^{-\lambda x}\)

\(\implies\) \(-ln(0.05) = \lambda x\)

\(\implies\) $ x = $

percent_5th <- round((-log(0.95)/lambda_fit),4)

cat("The 5th Percentile is given as : ", "\n", (percent_5th))
## The 5th Percentile is given as :  
##  539.4428
percent_95th <- round((-log(0.05)/lambda_fit),4)

cat("The 95th Percentile is given as : ", "\n", (percent_95th))
## The 95th Percentile is given as :  
##  31505.6
##    To Compute 95% confidence interval from the empirical data

mean_la_fit <-mean(la_fit)

p2_norm<-rnorm(length(la_fit),mean(la_fit),sd(la_fit))

cat("The 95th confidence interval from the data is given as : ", "\n",(quantile(p2_norm, probs=c(0.05, 0.95))))
## The 95th confidence interval from the data is given as :  
##  -5525.364 27175.39
#  The Histogram of the distribution is :

hist(p2_norm)

## The empirical 5th percentile and 95th percentile of the data is given as :

quantile(la_fit, c(0.05, 0.95))
##       5%      95% 
##  3311.70 17401.15

Modeling

Build some type of multiple regression model and submit your model to the competition board. Provide your complete model summary and results with analysis

## Overview of datasets including the addition converted categorical columns.

# Printing the enhanced data frame with new quantitative columns

# ------ Training Dataset

p2_train %>% select(order(colnames(p2_train)))
str(p2_train)
## 'data.frame':    1460 obs. of  88 variables:
##  $ Id            : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ MSSubClass    : int  60 20 60 70 60 50 20 60 50 190 ...
##  $ MSZoning      : chr  "RL" "RL" "RL" "RL" ...
##  $ LotFrontage   : num  65 80 68 60 84 85 75 0 51 50 ...
##  $ LotArea       : int  8450 9600 11250 9550 14260 14115 10084 10382 6120 7420 ...
##  $ Street        : chr  "Pave" "Pave" "Pave" "Pave" ...
##  $ Alley         : chr  "0" "0" "0" "0" ...
##  $ LotShape      : chr  "Reg" "Reg" "IR1" "IR1" ...
##  $ LandContour   : chr  "Lvl" "Lvl" "Lvl" "Lvl" ...
##  $ Utilities     : chr  "AllPub" "AllPub" "AllPub" "AllPub" ...
##  $ LotConfig     : chr  "Inside" "FR2" "Inside" "Corner" ...
##  $ LandSlope     : chr  "Gtl" "Gtl" "Gtl" "Gtl" ...
##  $ Neighborhood  : chr  "CollgCr" "Veenker" "CollgCr" "Crawfor" ...
##  $ Condition1    : chr  "Norm" "Feedr" "Norm" "Norm" ...
##  $ Condition2    : chr  "Norm" "Norm" "Norm" "Norm" ...
##  $ BldgType      : chr  "1Fam" "1Fam" "1Fam" "1Fam" ...
##  $ HouseStyle    : chr  "2Story" "1Story" "2Story" "2Story" ...
##  $ OverallQual   : int  7 6 7 7 8 5 8 7 7 5 ...
##  $ OverallCond   : int  5 8 5 5 5 5 5 6 5 6 ...
##  $ YearBuilt     : int  2003 1976 2001 1915 2000 1993 2004 1973 1931 1939 ...
##  $ YearRemodAdd  : int  2003 1976 2002 1970 2000 1995 2005 1973 1950 1950 ...
##  $ RoofStyle     : chr  "Gable" "Gable" "Gable" "Gable" ...
##  $ RoofMatl      : chr  "CompShg" "CompShg" "CompShg" "CompShg" ...
##  $ Exterior1st   : chr  "VinylSd" "MetalSd" "VinylSd" "Wd Sdng" ...
##  $ Exterior2nd   : chr  "VinylSd" "MetalSd" "VinylSd" "Wd Shng" ...
##  $ MasVnrType    : chr  "BrkFace" "None" "BrkFace" "None" ...
##  $ MasVnrArea    : num  196 0 162 0 350 0 186 240 0 0 ...
##  $ ExterQual     : chr  "Gd" "TA" "Gd" "TA" ...
##  $ ExterCond     : chr  "TA" "TA" "TA" "TA" ...
##  $ Foundation    : chr  "PConc" "CBlock" "PConc" "BrkTil" ...
##  $ BsmtQual      : chr  "Gd" "Gd" "Gd" "TA" ...
##  $ BsmtCond      : chr  "TA" "TA" "TA" "Gd" ...
##  $ BsmtExposure  : chr  "No" "Gd" "Mn" "No" ...
##  $ BsmtFinType1  : chr  "GLQ" "ALQ" "GLQ" "ALQ" ...
##  $ BsmtFinSF1    : int  706 978 486 216 655 732 1369 859 0 851 ...
##  $ BsmtFinType2  : chr  "Unf" "Unf" "Unf" "Unf" ...
##  $ BsmtFinSF2    : int  0 0 0 0 0 0 0 32 0 0 ...
##  $ BsmtUnfSF     : int  150 284 434 540 490 64 317 216 952 140 ...
##  $ TotalBsmtSF   : int  856 1262 920 756 1145 796 1686 1107 952 991 ...
##  $ Heating       : chr  "GasA" "GasA" "GasA" "GasA" ...
##  $ HeatingQC     : chr  "Ex" "Ex" "Ex" "Gd" ...
##  $ CentralAir    : chr  "Y" "Y" "Y" "Y" ...
##  $ Electrical    : chr  "SBrkr" "SBrkr" "SBrkr" "SBrkr" ...
##  $ X1stFlrSF     : int  856 1262 920 961 1145 796 1694 1107 1022 1077 ...
##  $ X2ndFlrSF     : int  854 0 866 756 1053 566 0 983 752 0 ...
##  $ LowQualFinSF  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ GrLivArea     : int  1710 1262 1786 1717 2198 1362 1694 2090 1774 1077 ...
##  $ BsmtFullBath  : int  1 0 1 1 1 1 1 1 0 1 ...
##  $ BsmtHalfBath  : int  0 1 0 0 0 0 0 0 0 0 ...
##  $ FullBath      : int  2 2 2 1 2 1 2 2 2 1 ...
##  $ HalfBath      : int  1 0 1 0 1 1 0 1 0 0 ...
##  $ BedroomAbvGr  : int  3 3 3 3 4 1 3 3 2 2 ...
##  $ KitchenAbvGr  : int  1 1 1 1 1 1 1 1 2 2 ...
##  $ KitchenQual   : chr  "Gd" "TA" "Gd" "Gd" ...
##  $ TotRmsAbvGrd  : int  8 6 6 7 9 5 7 7 8 5 ...
##  $ Functional    : chr  "Typ" "Typ" "Typ" "Typ" ...
##  $ Fireplaces    : int  0 1 1 1 1 0 1 2 2 2 ...
##  $ FireplaceQu   : chr  "0" "TA" "TA" "Gd" ...
##  $ GarageType    : chr  "Attchd" "Attchd" "Attchd" "Detchd" ...
##  $ GarageYrBlt   : num  2003 1976 2001 1998 2000 ...
##  $ GarageFinish  : chr  "RFn" "RFn" "RFn" "Unf" ...
##  $ GarageCars    : int  2 2 2 3 3 2 2 2 2 1 ...
##  $ GarageArea    : int  548 460 608 642 836 480 636 484 468 205 ...
##  $ GarageQual    : chr  "TA" "TA" "TA" "TA" ...
##  $ GarageCond    : chr  "TA" "TA" "TA" "TA" ...
##  $ PavedDrive    : chr  "Y" "Y" "Y" "Y" ...
##  $ WoodDeckSF    : int  0 298 0 0 192 40 255 235 90 0 ...
##  $ OpenPorchSF   : int  61 0 42 35 84 30 57 204 0 4 ...
##  $ EnclosedPorch : int  0 0 0 272 0 0 0 228 205 0 ...
##  $ X3SsnPorch    : int  0 0 0 0 0 320 0 0 0 0 ...
##  $ ScreenPorch   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolArea      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolQC        : chr  "0" "0" "0" "0" ...
##  $ Fence         : chr  "0" "0" "0" "0" ...
##  $ MiscFeature   : chr  "0" "0" "0" "0" ...
##  $ MiscVal       : int  0 0 0 0 0 700 0 350 0 0 ...
##  $ MoSold        : int  2 5 9 2 12 10 8 11 4 1 ...
##  $ YrSold        : int  2008 2007 2008 2006 2008 2009 2007 2009 2008 2008 ...
##  $ SaleType      : chr  "WD" "WD" "WD" "WD" ...
##  $ SaleCondition : chr  "Normal" "Normal" "Normal" "Abnorml" ...
##  $ SalePrice     : int  208500 181500 223500 140000 250000 143000 307000 200000 129900 118000 ...
##  $ Foundation_q  : num  4 5 4 6 4 1 4 5 6 6 ...
##  $ BsmtFinType2_q: num  1 1 1 1 1 1 1 4 1 1 ...
##  $ HeatingQC_q   : num  5 5 5 4 5 5 5 5 4 5 ...
##  $ Electrical_q  : num  5 5 5 5 5 5 5 5 3 5 ...
##  $ KitchenQual_q : num  4 3 4 4 4 3 4 3 3 3 ...
##  $ GarageCond_q  : num  3 3 3 3 3 3 3 3 3 3 ...
##  $ Fence_q       : num  0 0 0 0 0 3 0 0 0 0 ...
kable(head(p2_train))
Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape LandContour Utilities LotConfig LandSlope Neighborhood Condition1 Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold SaleType SaleCondition SalePrice Foundation_q BsmtFinType2_q HeatingQC_q Electrical_q KitchenQual_q GarageCond_q Fence_q
1 60 RL 65 8450 Pave 0 Reg Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2003 2003 Gable CompShg VinylSd VinylSd BrkFace 196 Gd TA PConc Gd TA No GLQ 706 Unf 0 150 856 GasA Ex Y SBrkr 856 854 0 1710 1 0 2 1 3 1 Gd 8 Typ 0 0 Attchd 2003 RFn 2 548 TA TA Y 0 61 0 0 0 0 0 0 0 0 2 2008 WD Normal 208500 4 1 5 5 4 3 0
2 20 RL 80 9600 Pave 0 Reg Lvl AllPub FR2 Gtl Veenker Feedr Norm 1Fam 1Story 6 8 1976 1976 Gable CompShg MetalSd MetalSd None 0 TA TA CBlock Gd TA Gd ALQ 978 Unf 0 284 1262 GasA Ex Y SBrkr 1262 0 0 1262 0 1 2 0 3 1 TA 6 Typ 1 TA Attchd 1976 RFn 2 460 TA TA Y 298 0 0 0 0 0 0 0 0 0 5 2007 WD Normal 181500 5 1 5 5 3 3 0
3 60 RL 68 11250 Pave 0 IR1 Lvl AllPub Inside Gtl CollgCr Norm Norm 1Fam 2Story 7 5 2001 2002 Gable CompShg VinylSd VinylSd BrkFace 162 Gd TA PConc Gd TA Mn GLQ 486 Unf 0 434 920 GasA Ex Y SBrkr 920 866 0 1786 1 0 2 1 3 1 Gd 6 Typ 1 TA Attchd 2001 RFn 2 608 TA TA Y 0 42 0 0 0 0 0 0 0 0 9 2008 WD Normal 223500 4 1 5 5 4 3 0
4 70 RL 60 9550 Pave 0 IR1 Lvl AllPub Corner Gtl Crawfor Norm Norm 1Fam 2Story 7 5 1915 1970 Gable CompShg Wd Sdng Wd Shng None 0 TA TA BrkTil TA Gd No ALQ 216 Unf 0 540 756 GasA Gd Y SBrkr 961 756 0 1717 1 0 1 0 3 1 Gd 7 Typ 1 Gd Detchd 1998 Unf 3 642 TA TA Y 0 35 272 0 0 0 0 0 0 0 2 2006 WD Abnorml 140000 6 1 4 5 4 3 0
5 60 RL 84 14260 Pave 0 IR1 Lvl AllPub FR2 Gtl NoRidge Norm Norm 1Fam 2Story 8 5 2000 2000 Gable CompShg VinylSd VinylSd BrkFace 350 Gd TA PConc Gd TA Av GLQ 655 Unf 0 490 1145 GasA Ex Y SBrkr 1145 1053 0 2198 1 0 2 1 4 1 Gd 9 Typ 1 TA Attchd 2000 RFn 3 836 TA TA Y 192 84 0 0 0 0 0 0 0 0 12 2008 WD Normal 250000 4 1 5 5 4 3 0
6 50 RL 85 14115 Pave 0 IR1 Lvl AllPub Inside Gtl Mitchel Norm Norm 1Fam 1.5Fin 5 5 1993 1995 Gable CompShg VinylSd VinylSd None 0 TA TA Wood Gd TA No GLQ 732 Unf 0 64 796 GasA Ex Y SBrkr 796 566 0 1362 1 0 1 1 1 1 TA 5 Typ 0 0 Attchd 1993 Unf 2 480 TA TA Y 40 30 0 320 0 0 0 MnPrv Shed 700 10 2009 WD Normal 143000 1 1 5 5 3 3 3
# ------ Test Dataset

p2_test %>% select(order(colnames(p2_test)))
str(p2_test)
## 'data.frame':    1459 obs. of  88 variables:
##  $ Id            : int  1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 ...
##  $ MSSubClass    : int  20 20 60 60 120 60 20 60 20 20 ...
##  $ MSZoning      : chr  "RH" "RL" "RL" "RL" ...
##  $ LotFrontage   : num  80 81 74 78 43 75 0 63 85 70 ...
##  $ LotArea       : int  11622 14267 13830 9978 5005 10000 7980 8402 10176 8400 ...
##  $ Street        : chr  "Pave" "Pave" "Pave" "Pave" ...
##  $ Alley         : chr  "0" "0" "0" "0" ...
##  $ LotShape      : chr  "Reg" "IR1" "IR1" "IR1" ...
##  $ LandContour   : chr  "Lvl" "Lvl" "Lvl" "Lvl" ...
##  $ Utilities     : chr  "AllPub" "AllPub" "AllPub" "AllPub" ...
##  $ LotConfig     : chr  "Inside" "Corner" "Inside" "Inside" ...
##  $ LandSlope     : chr  "Gtl" "Gtl" "Gtl" "Gtl" ...
##  $ Neighborhood  : chr  "NAmes" "NAmes" "Gilbert" "Gilbert" ...
##  $ Condition1    : chr  "Feedr" "Norm" "Norm" "Norm" ...
##  $ Condition2    : chr  "Norm" "Norm" "Norm" "Norm" ...
##  $ BldgType      : chr  "1Fam" "1Fam" "1Fam" "1Fam" ...
##  $ HouseStyle    : chr  "1Story" "1Story" "2Story" "2Story" ...
##  $ OverallQual   : int  5 6 5 6 8 6 6 6 7 4 ...
##  $ OverallCond   : int  6 6 5 6 5 5 7 5 5 5 ...
##  $ YearBuilt     : int  1961 1958 1997 1998 1992 1993 1992 1998 1990 1970 ...
##  $ YearRemodAdd  : int  1961 1958 1998 1998 1992 1994 2007 1998 1990 1970 ...
##  $ RoofStyle     : chr  "Gable" "Hip" "Gable" "Gable" ...
##  $ RoofMatl      : chr  "CompShg" "CompShg" "CompShg" "CompShg" ...
##  $ Exterior1st   : chr  "VinylSd" "Wd Sdng" "VinylSd" "VinylSd" ...
##  $ Exterior2nd   : chr  "VinylSd" "Wd Sdng" "VinylSd" "VinylSd" ...
##  $ MasVnrType    : chr  "None" "BrkFace" "None" "BrkFace" ...
##  $ MasVnrArea    : num  0 108 0 20 0 0 0 0 0 0 ...
##  $ ExterQual     : chr  "TA" "TA" "TA" "TA" ...
##  $ ExterCond     : chr  "TA" "TA" "TA" "TA" ...
##  $ Foundation    : chr  "CBlock" "CBlock" "PConc" "PConc" ...
##  $ BsmtQual      : chr  "TA" "TA" "Gd" "TA" ...
##  $ BsmtCond      : chr  "TA" "TA" "TA" "TA" ...
##  $ BsmtExposure  : chr  "No" "No" "No" "No" ...
##  $ BsmtFinType1  : chr  "Rec" "ALQ" "GLQ" "GLQ" ...
##  $ BsmtFinSF1    : num  468 923 791 602 263 0 935 0 637 804 ...
##  $ BsmtFinType2  : chr  "LwQ" "Unf" "Unf" "Unf" ...
##  $ BsmtFinSF2    : num  144 0 0 0 0 0 0 0 0 78 ...
##  $ BsmtUnfSF     : num  270 406 137 324 1017 ...
##  $ TotalBsmtSF   : num  882 1329 928 926 1280 ...
##  $ Heating       : chr  "GasA" "GasA" "GasA" "GasA" ...
##  $ HeatingQC     : chr  "TA" "TA" "Gd" "Ex" ...
##  $ CentralAir    : chr  "Y" "Y" "Y" "Y" ...
##  $ Electrical    : chr  "SBrkr" "SBrkr" "SBrkr" "SBrkr" ...
##  $ X1stFlrSF     : int  896 1329 928 926 1280 763 1187 789 1341 882 ...
##  $ X2ndFlrSF     : int  0 0 701 678 0 892 0 676 0 0 ...
##  $ LowQualFinSF  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ GrLivArea     : int  896 1329 1629 1604 1280 1655 1187 1465 1341 882 ...
##  $ BsmtFullBath  : num  0 0 0 0 0 0 1 0 1 1 ...
##  $ BsmtHalfBath  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ FullBath      : int  1 1 2 2 2 2 2 2 1 1 ...
##  $ HalfBath      : int  0 1 1 1 0 1 0 1 1 0 ...
##  $ BedroomAbvGr  : int  2 3 3 3 2 3 3 3 2 2 ...
##  $ KitchenAbvGr  : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ KitchenQual   : chr  "TA" "Gd" "TA" "Gd" ...
##  $ TotRmsAbvGrd  : int  5 6 6 7 5 7 6 7 5 4 ...
##  $ Functional    : chr  "Typ" "Typ" "Typ" "Typ" ...
##  $ Fireplaces    : int  0 0 1 1 0 1 0 1 1 0 ...
##  $ FireplaceQu   : chr  "0" "0" "TA" "Gd" ...
##  $ GarageType    : chr  "Attchd" "Attchd" "Attchd" "Attchd" ...
##  $ GarageYrBlt   : num  1961 1958 1997 1998 1992 ...
##  $ GarageFinish  : chr  "Unf" "Unf" "Fin" "Fin" ...
##  $ GarageCars    : num  1 1 2 2 2 2 2 2 2 2 ...
##  $ GarageArea    : num  730 312 482 470 506 440 420 393 506 525 ...
##  $ GarageQual    : chr  "TA" "TA" "TA" "TA" ...
##  $ GarageCond    : chr  "TA" "TA" "TA" "TA" ...
##  $ PavedDrive    : chr  "Y" "Y" "Y" "Y" ...
##  $ WoodDeckSF    : int  140 393 212 360 0 157 483 0 192 240 ...
##  $ OpenPorchSF   : int  0 36 34 36 82 84 21 75 0 0 ...
##  $ EnclosedPorch : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ X3SsnPorch    : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ScreenPorch   : int  120 0 0 0 144 0 0 0 0 0 ...
##  $ PoolArea      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolQC        : chr  "0" "0" "0" "0" ...
##  $ Fence         : chr  "MnPrv" "0" "MnPrv" "0" ...
##  $ MiscFeature   : chr  "0" "Gar2" "0" "0" ...
##  $ MiscVal       : int  0 12500 0 0 0 0 500 0 0 0 ...
##  $ MoSold        : int  6 6 3 6 1 4 3 5 2 4 ...
##  $ YrSold        : int  2010 2010 2010 2010 2010 2010 2010 2010 2010 2010 ...
##  $ SaleType      : chr  "WD" "WD" "WD" "WD" ...
##  $ SaleCondition : chr  "Normal" "Normal" "Normal" "Normal" ...
##  $ SalePrice     : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Foundation_q  : num  5 5 4 4 4 4 4 4 4 5 ...
##  $ BsmtFinType2_q: num  2 1 1 1 1 1 1 1 1 3 ...
##  $ HeatingQC_q   : num  3 3 4 5 5 4 5 4 4 3 ...
##  $ Electrical_q  : num  5 5 5 5 5 5 5 5 5 5 ...
##  $ KitchenQual_q : num  3 4 3 4 4 3 3 3 4 3 ...
##  $ GarageCond_q  : num  3 3 3 3 3 3 3 3 3 3 ...
##  $ Fence_q       : num  3 0 3 0 0 0 4 0 0 3 ...
kable(head(p2_test))
Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape LandContour Utilities LotConfig LandSlope Neighborhood Condition1 Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold SaleType SaleCondition SalePrice Foundation_q BsmtFinType2_q HeatingQC_q Electrical_q KitchenQual_q GarageCond_q Fence_q
1461 20 RH 80 11622 Pave 0 Reg Lvl AllPub Inside Gtl NAmes Feedr Norm 1Fam 1Story 5 6 1961 1961 Gable CompShg VinylSd VinylSd None 0 TA TA CBlock TA TA No Rec 468 LwQ 144 270 882 GasA TA Y SBrkr 896 0 0 896 0 0 1 0 2 1 TA 5 Typ 0 0 Attchd 1961 Unf 1 730 TA TA Y 140 0 0 0 120 0 0 MnPrv 0 0 6 2010 WD Normal 0 5 2 3 5 3 3 3
1462 20 RL 81 14267 Pave 0 IR1 Lvl AllPub Corner Gtl NAmes Norm Norm 1Fam 1Story 6 6 1958 1958 Hip CompShg Wd Sdng Wd Sdng BrkFace 108 TA TA CBlock TA TA No ALQ 923 Unf 0 406 1329 GasA TA Y SBrkr 1329 0 0 1329 0 0 1 1 3 1 Gd 6 Typ 0 0 Attchd 1958 Unf 1 312 TA TA Y 393 36 0 0 0 0 0 0 Gar2 12500 6 2010 WD Normal 0 5 1 3 5 4 3 0
1463 60 RL 74 13830 Pave 0 IR1 Lvl AllPub Inside Gtl Gilbert Norm Norm 1Fam 2Story 5 5 1997 1998 Gable CompShg VinylSd VinylSd None 0 TA TA PConc Gd TA No GLQ 791 Unf 0 137 928 GasA Gd Y SBrkr 928 701 0 1629 0 0 2 1 3 1 TA 6 Typ 1 TA Attchd 1997 Fin 2 482 TA TA Y 212 34 0 0 0 0 0 MnPrv 0 0 3 2010 WD Normal 0 4 1 4 5 3 3 3
1464 60 RL 78 9978 Pave 0 IR1 Lvl AllPub Inside Gtl Gilbert Norm Norm 1Fam 2Story 6 6 1998 1998 Gable CompShg VinylSd VinylSd BrkFace 20 TA TA PConc TA TA No GLQ 602 Unf 0 324 926 GasA Ex Y SBrkr 926 678 0 1604 0 0 2 1 3 1 Gd 7 Typ 1 Gd Attchd 1998 Fin 2 470 TA TA Y 360 36 0 0 0 0 0 0 0 0 6 2010 WD Normal 0 4 1 5 5 4 3 0
1465 120 RL 43 5005 Pave 0 IR1 HLS AllPub Inside Gtl StoneBr Norm Norm TwnhsE 1Story 8 5 1992 1992 Gable CompShg HdBoard HdBoard None 0 Gd TA PConc Gd TA No ALQ 263 Unf 0 1017 1280 GasA Ex Y SBrkr 1280 0 0 1280 0 0 2 0 2 1 Gd 5 Typ 0 0 Attchd 1992 RFn 2 506 TA TA Y 0 82 0 0 144 0 0 0 0 0 1 2010 WD Normal 0 4 1 5 5 4 3 0
1466 60 RL 75 10000 Pave 0 IR1 Lvl AllPub Corner Gtl Gilbert Norm Norm 1Fam 2Story 6 5 1993 1994 Gable CompShg HdBoard HdBoard None 0 TA TA PConc Gd TA No Unf 0 Unf 0 763 763 GasA Gd Y SBrkr 763 892 0 1655 0 0 2 1 3 1 TA 7 Typ 1 TA Attchd 1993 Fin 2 440 TA TA Y 157 84 0 0 0 0 0 0 0 0 4 2010 WD Normal 0 4 1 4 5 3 3 0

Preparing the Training dataset by removing the non-numerical columns

# selection columns that are numeric only
p2_train_num <- p2_train %>% 
  dplyr::select_if(is.numeric)

# Dropping the "id" and "SalePrice" fields since it is not needed for the predictor model Variables

p2_train_vars <- subset(p2_train_num, select = -c(Id,SalePrice))

# Check for missing values in data 

colSums(is.na(p2_train_vars))
##     MSSubClass    LotFrontage        LotArea    OverallQual    OverallCond 
##              0              0              0              0              0 
##      YearBuilt   YearRemodAdd     MasVnrArea     BsmtFinSF1     BsmtFinSF2 
##              0              0              0              0              0 
##      BsmtUnfSF    TotalBsmtSF      X1stFlrSF      X2ndFlrSF   LowQualFinSF 
##              0              0              0              0              0 
##      GrLivArea   BsmtFullBath   BsmtHalfBath       FullBath       HalfBath 
##              0              0              0              0              0 
##   BedroomAbvGr   KitchenAbvGr   TotRmsAbvGrd     Fireplaces    GarageYrBlt 
##              0              0              0              0              0 
##     GarageCars     GarageArea     WoodDeckSF    OpenPorchSF  EnclosedPorch 
##              0              0              0              0              0 
##     X3SsnPorch    ScreenPorch       PoolArea        MiscVal         MoSold 
##              0              0              0              0              0 
##         YrSold   Foundation_q BsmtFinType2_q    HeatingQC_q   Electrical_q 
##              0              0              0              0              0 
##  KitchenQual_q   GarageCond_q        Fence_q 
##              0              0              0
## Reviewing the structure of the enhanced dataset

str(p2_train_vars)
## 'data.frame':    1460 obs. of  43 variables:
##  $ MSSubClass    : int  60 20 60 70 60 50 20 60 50 190 ...
##  $ LotFrontage   : num  65 80 68 60 84 85 75 0 51 50 ...
##  $ LotArea       : int  8450 9600 11250 9550 14260 14115 10084 10382 6120 7420 ...
##  $ OverallQual   : int  7 6 7 7 8 5 8 7 7 5 ...
##  $ OverallCond   : int  5 8 5 5 5 5 5 6 5 6 ...
##  $ YearBuilt     : int  2003 1976 2001 1915 2000 1993 2004 1973 1931 1939 ...
##  $ YearRemodAdd  : int  2003 1976 2002 1970 2000 1995 2005 1973 1950 1950 ...
##  $ MasVnrArea    : num  196 0 162 0 350 0 186 240 0 0 ...
##  $ BsmtFinSF1    : int  706 978 486 216 655 732 1369 859 0 851 ...
##  $ BsmtFinSF2    : int  0 0 0 0 0 0 0 32 0 0 ...
##  $ BsmtUnfSF     : int  150 284 434 540 490 64 317 216 952 140 ...
##  $ TotalBsmtSF   : int  856 1262 920 756 1145 796 1686 1107 952 991 ...
##  $ X1stFlrSF     : int  856 1262 920 961 1145 796 1694 1107 1022 1077 ...
##  $ X2ndFlrSF     : int  854 0 866 756 1053 566 0 983 752 0 ...
##  $ LowQualFinSF  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ GrLivArea     : int  1710 1262 1786 1717 2198 1362 1694 2090 1774 1077 ...
##  $ BsmtFullBath  : int  1 0 1 1 1 1 1 1 0 1 ...
##  $ BsmtHalfBath  : int  0 1 0 0 0 0 0 0 0 0 ...
##  $ FullBath      : int  2 2 2 1 2 1 2 2 2 1 ...
##  $ HalfBath      : int  1 0 1 0 1 1 0 1 0 0 ...
##  $ BedroomAbvGr  : int  3 3 3 3 4 1 3 3 2 2 ...
##  $ KitchenAbvGr  : int  1 1 1 1 1 1 1 1 2 2 ...
##  $ TotRmsAbvGrd  : int  8 6 6 7 9 5 7 7 8 5 ...
##  $ Fireplaces    : int  0 1 1 1 1 0 1 2 2 2 ...
##  $ GarageYrBlt   : num  2003 1976 2001 1998 2000 ...
##  $ GarageCars    : int  2 2 2 3 3 2 2 2 2 1 ...
##  $ GarageArea    : int  548 460 608 642 836 480 636 484 468 205 ...
##  $ WoodDeckSF    : int  0 298 0 0 192 40 255 235 90 0 ...
##  $ OpenPorchSF   : int  61 0 42 35 84 30 57 204 0 4 ...
##  $ EnclosedPorch : int  0 0 0 272 0 0 0 228 205 0 ...
##  $ X3SsnPorch    : int  0 0 0 0 0 320 0 0 0 0 ...
##  $ ScreenPorch   : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolArea      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ MiscVal       : int  0 0 0 0 0 700 0 350 0 0 ...
##  $ MoSold        : int  2 5 9 2 12 10 8 11 4 1 ...
##  $ YrSold        : int  2008 2007 2008 2006 2008 2009 2007 2009 2008 2008 ...
##  $ Foundation_q  : num  4 5 4 6 4 1 4 5 6 6 ...
##  $ BsmtFinType2_q: num  1 1 1 1 1 1 1 4 1 1 ...
##  $ HeatingQC_q   : num  5 5 5 4 5 5 5 5 4 5 ...
##  $ Electrical_q  : num  5 5 5 5 5 5 5 5 3 5 ...
##  $ KitchenQual_q : num  4 3 4 4 4 3 4 3 3 3 ...
##  $ GarageCond_q  : num  3 3 3 3 3 3 3 3 3 3 ...
##  $ Fence_q       : num  0 0 0 0 0 3 0 0 0 0 ...
dim(p2_train_vars)
## [1] 1460   43
kable(head(p2_train_vars))
MSSubClass LotFrontage LotArea OverallQual OverallCond YearBuilt YearRemodAdd MasVnrArea BsmtFinSF1 BsmtFinSF2 BsmtUnfSF TotalBsmtSF X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr TotRmsAbvGrd Fireplaces GarageYrBlt GarageCars GarageArea WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea MiscVal MoSold YrSold Foundation_q BsmtFinType2_q HeatingQC_q Electrical_q KitchenQual_q GarageCond_q Fence_q
60 65 8450 7 5 2003 2003 196 706 0 150 856 856 854 0 1710 1 0 2 1 3 1 8 0 2003 2 548 0 61 0 0 0 0 0 2 2008 4 1 5 5 4 3 0
20 80 9600 6 8 1976 1976 0 978 0 284 1262 1262 0 0 1262 0 1 2 0 3 1 6 1 1976 2 460 298 0 0 0 0 0 0 5 2007 5 1 5 5 3 3 0
60 68 11250 7 5 2001 2002 162 486 0 434 920 920 866 0 1786 1 0 2 1 3 1 6 1 2001 2 608 0 42 0 0 0 0 0 9 2008 4 1 5 5 4 3 0
70 60 9550 7 5 1915 1970 0 216 0 540 756 961 756 0 1717 1 0 1 0 3 1 7 1 1998 3 642 0 35 272 0 0 0 0 2 2006 6 1 4 5 4 3 0
60 84 14260 8 5 2000 2000 350 655 0 490 1145 1145 1053 0 2198 1 0 2 1 4 1 9 1 2000 3 836 192 84 0 0 0 0 0 12 2008 4 1 5 5 4 3 0
50 85 14115 5 5 1993 1995 0 732 0 64 796 796 566 0 1362 1 0 1 1 1 1 5 0 1993 2 480 40 30 0 320 0 0 700 10 2009 1 1 5 5 3 3 3

Since our initial data preparation yielded 43 numerical variables that are eligible to be included in the linear model, we will use additional tools to narrow the selection to variables that will yield “best fit” results. The two computations to be employed are :

Test for Multicollinearity

Test for Correlation to the predicted variable - Sales Price -

##  the summary of the dataset is : 

summary(p2_train_vars)
##    MSSubClass     LotFrontage        LotArea        OverallQual    
##  Min.   : 20.0   Min.   :  0.00   Min.   :  1300   Min.   : 1.000  
##  1st Qu.: 20.0   1st Qu.: 42.00   1st Qu.:  7554   1st Qu.: 5.000  
##  Median : 50.0   Median : 63.00   Median :  9478   Median : 6.000  
##  Mean   : 56.9   Mean   : 57.62   Mean   : 10517   Mean   : 6.099  
##  3rd Qu.: 70.0   3rd Qu.: 79.00   3rd Qu.: 11602   3rd Qu.: 7.000  
##  Max.   :190.0   Max.   :313.00   Max.   :215245   Max.   :10.000  
##   OverallCond      YearBuilt     YearRemodAdd    MasVnrArea    
##  Min.   :1.000   Min.   :1872   Min.   :1950   Min.   :   0.0  
##  1st Qu.:5.000   1st Qu.:1954   1st Qu.:1967   1st Qu.:   0.0  
##  Median :5.000   Median :1973   Median :1994   Median :   0.0  
##  Mean   :5.575   Mean   :1971   Mean   :1985   Mean   : 103.1  
##  3rd Qu.:6.000   3rd Qu.:2000   3rd Qu.:2004   3rd Qu.: 164.2  
##  Max.   :9.000   Max.   :2010   Max.   :2010   Max.   :1600.0  
##    BsmtFinSF1       BsmtFinSF2        BsmtUnfSF       TotalBsmtSF    
##  Min.   :   0.0   Min.   :   0.00   Min.   :   0.0   Min.   :   0.0  
##  1st Qu.:   0.0   1st Qu.:   0.00   1st Qu.: 223.0   1st Qu.: 795.8  
##  Median : 383.5   Median :   0.00   Median : 477.5   Median : 991.5  
##  Mean   : 443.6   Mean   :  46.55   Mean   : 567.2   Mean   :1057.4  
##  3rd Qu.: 712.2   3rd Qu.:   0.00   3rd Qu.: 808.0   3rd Qu.:1298.2  
##  Max.   :5644.0   Max.   :1474.00   Max.   :2336.0   Max.   :6110.0  
##    X1stFlrSF      X2ndFlrSF     LowQualFinSF       GrLivArea   
##  Min.   : 334   Min.   :   0   Min.   :  0.000   Min.   : 334  
##  1st Qu.: 882   1st Qu.:   0   1st Qu.:  0.000   1st Qu.:1130  
##  Median :1087   Median :   0   Median :  0.000   Median :1464  
##  Mean   :1163   Mean   : 347   Mean   :  5.845   Mean   :1515  
##  3rd Qu.:1391   3rd Qu.: 728   3rd Qu.:  0.000   3rd Qu.:1777  
##  Max.   :4692   Max.   :2065   Max.   :572.000   Max.   :5642  
##   BsmtFullBath     BsmtHalfBath        FullBath        HalfBath     
##  Min.   :0.0000   Min.   :0.00000   Min.   :0.000   Min.   :0.0000  
##  1st Qu.:0.0000   1st Qu.:0.00000   1st Qu.:1.000   1st Qu.:0.0000  
##  Median :0.0000   Median :0.00000   Median :2.000   Median :0.0000  
##  Mean   :0.4253   Mean   :0.05753   Mean   :1.565   Mean   :0.3829  
##  3rd Qu.:1.0000   3rd Qu.:0.00000   3rd Qu.:2.000   3rd Qu.:1.0000  
##  Max.   :3.0000   Max.   :2.00000   Max.   :3.000   Max.   :2.0000  
##   BedroomAbvGr    KitchenAbvGr    TotRmsAbvGrd      Fireplaces   
##  Min.   :0.000   Min.   :0.000   Min.   : 2.000   Min.   :0.000  
##  1st Qu.:2.000   1st Qu.:1.000   1st Qu.: 5.000   1st Qu.:0.000  
##  Median :3.000   Median :1.000   Median : 6.000   Median :1.000  
##  Mean   :2.866   Mean   :1.047   Mean   : 6.518   Mean   :0.613  
##  3rd Qu.:3.000   3rd Qu.:1.000   3rd Qu.: 7.000   3rd Qu.:1.000  
##  Max.   :8.000   Max.   :3.000   Max.   :14.000   Max.   :3.000  
##   GarageYrBlt     GarageCars      GarageArea       WoodDeckSF    
##  Min.   :   0   Min.   :0.000   Min.   :   0.0   Min.   :  0.00  
##  1st Qu.:1958   1st Qu.:1.000   1st Qu.: 334.5   1st Qu.:  0.00  
##  Median :1977   Median :2.000   Median : 480.0   Median :  0.00  
##  Mean   :1869   Mean   :1.767   Mean   : 473.0   Mean   : 94.24  
##  3rd Qu.:2001   3rd Qu.:2.000   3rd Qu.: 576.0   3rd Qu.:168.00  
##  Max.   :2010   Max.   :4.000   Max.   :1418.0   Max.   :857.00  
##   OpenPorchSF     EnclosedPorch      X3SsnPorch      ScreenPorch    
##  Min.   :  0.00   Min.   :  0.00   Min.   :  0.00   Min.   :  0.00  
##  1st Qu.:  0.00   1st Qu.:  0.00   1st Qu.:  0.00   1st Qu.:  0.00  
##  Median : 25.00   Median :  0.00   Median :  0.00   Median :  0.00  
##  Mean   : 46.66   Mean   : 21.95   Mean   :  3.41   Mean   : 15.06  
##  3rd Qu.: 68.00   3rd Qu.:  0.00   3rd Qu.:  0.00   3rd Qu.:  0.00  
##  Max.   :547.00   Max.   :552.00   Max.   :508.00   Max.   :480.00  
##     PoolArea          MiscVal             MoSold           YrSold    
##  Min.   :  0.000   Min.   :    0.00   Min.   : 1.000   Min.   :2006  
##  1st Qu.:  0.000   1st Qu.:    0.00   1st Qu.: 5.000   1st Qu.:2007  
##  Median :  0.000   Median :    0.00   Median : 6.000   Median :2008  
##  Mean   :  2.759   Mean   :   43.49   Mean   : 6.322   Mean   :2008  
##  3rd Qu.:  0.000   3rd Qu.:    0.00   3rd Qu.: 8.000   3rd Qu.:2009  
##  Max.   :738.000   Max.   :15500.00   Max.   :12.000   Max.   :2010  
##   Foundation_q   BsmtFinType2_q   HeatingQC_q     Electrical_q  
##  Min.   :1.000   Min.   :0.000   Min.   :1.000   Min.   :0.000  
##  1st Qu.:4.000   1st Qu.:1.000   1st Qu.:3.000   1st Qu.:5.000  
##  Median :5.000   Median :1.000   Median :5.000   Median :5.000  
##  Mean   :4.603   Mean   :1.247   Mean   :4.145   Mean   :4.886  
##  3rd Qu.:5.000   3rd Qu.:1.000   3rd Qu.:5.000   3rd Qu.:5.000  
##  Max.   :6.000   Max.   :6.000   Max.   :5.000   Max.   :5.000  
##  KitchenQual_q    GarageCond_q      Fence_q      
##  Min.   :2.000   Min.   :0.000   Min.   :0.0000  
##  1st Qu.:3.000   1st Qu.:3.000   1st Qu.:0.0000  
##  Median :3.000   Median :3.000   Median :0.0000  
##  Mean   :3.512   Mean   :2.809   Mean   :0.5658  
##  3rd Qu.:4.000   3rd Qu.:3.000   3rd Qu.:0.0000  
##  Max.   :5.000   Max.   :5.000   Max.   :4.0000
## Checking for Null Values 

p2_train_vars[!complete.cases(p2_train_vars),]
## Predictor Variables included in initial regression : 

sort(colnames(p2_train_vars))
##  [1] "BedroomAbvGr"   "BsmtFinSF1"     "BsmtFinSF2"     "BsmtFinType2_q"
##  [5] "BsmtFullBath"   "BsmtHalfBath"   "BsmtUnfSF"      "Electrical_q"  
##  [9] "EnclosedPorch"  "Fence_q"        "Fireplaces"     "Foundation_q"  
## [13] "FullBath"       "GarageArea"     "GarageCars"     "GarageCond_q"  
## [17] "GarageYrBlt"    "GrLivArea"      "HalfBath"       "HeatingQC_q"   
## [21] "KitchenAbvGr"   "KitchenQual_q"  "LotArea"        "LotFrontage"   
## [25] "LowQualFinSF"   "MasVnrArea"     "MiscVal"        "MoSold"        
## [29] "MSSubClass"     "OpenPorchSF"    "OverallCond"    "OverallQual"   
## [33] "PoolArea"       "ScreenPorch"    "TotalBsmtSF"    "TotRmsAbvGrd"  
## [37] "WoodDeckSF"     "X1stFlrSF"      "X2ndFlrSF"      "X3SsnPorch"    
## [41] "YearBuilt"      "YearRemodAdd"   "YrSold"

Regression Model Fitting

We will perform Regression Modeling and manipulate predictor variables to compute the optimal outcome

Regression Modeling V1 and V2

Note that after computing Linear Model v2, we have Multiple - \(R^{2} = 0.823\) and \(R^{2} = 0.8183\)

##### Note that object "p2_train_vars" holds all predictor variables- columns

p2_train_regr_v1 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars)), collapse = "+"),
        sep = ""
    ))

## The Resultant variable list :

p2_train_regr_v1
## SalePrice ~ BedroomAbvGr + BsmtFinSF1 + BsmtFinSF2 + BsmtFinType2_q + 
##     BsmtFullBath + BsmtHalfBath + BsmtUnfSF + Electrical_q + 
##     EnclosedPorch + Fence_q + Fireplaces + Foundation_q + FullBath + 
##     GarageArea + GarageCars + GarageCond_q + GarageYrBlt + GrLivArea + 
##     HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + 
##     LotFrontage + LowQualFinSF + MasVnrArea + MiscVal + MoSold + 
##     MSSubClass + OpenPorchSF + OverallCond + OverallQual + PoolArea + 
##     ScreenPorch + TotalBsmtSF + TotRmsAbvGrd + WoodDeckSF + X1stFlrSF + 
##     X2ndFlrSF + X3SsnPorch + YearBuilt + YearRemodAdd + YrSold
#--------------- Linear Model Version 1 --------------

lm_1 <- lm((p2_train_regr_v1),data = p2_train)

summary(lm_1)
## 
## Call:
## lm(formula = (p2_train_regr_v1), data = p2_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -472691  -16414   -1979   13200  297806 
## 
## Coefficients: (2 not defined because of singularities)
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     9.261e+05  1.385e+06   0.669 0.503705    
## BedroomAbvGr   -8.213e+03  1.683e+03  -4.881 1.17e-06 ***
## BsmtFinSF1      1.986e+01  4.669e+00   4.255 2.23e-05 ***
## BsmtFinSF2      1.577e+01  1.048e+01   1.505 0.132474    
## BsmtFinType2_q -7.895e+02  1.703e+03  -0.464 0.642987    
## BsmtFullBath    8.031e+03  2.567e+03   3.129 0.001792 ** 
## BsmtHalfBath    1.919e+03  3.998e+03   0.480 0.631350    
## BsmtUnfSF       9.120e+00  4.279e+00   2.131 0.033221 *  
## Electrical_q   -3.220e+03  2.401e+03  -1.341 0.180057    
## EnclosedPorch   2.458e+00  1.653e+01   0.149 0.881788    
## Fence_q        -1.052e+03  7.984e+02  -1.317 0.187955    
## Fireplaces      4.547e+03  1.734e+03   2.622 0.008841 ** 
## Foundation_q   -3.347e+03  1.714e+03  -1.953 0.051019 .  
## FullBath        3.184e+03  2.762e+03   1.153 0.249170    
## GarageArea      3.283e+00  9.608e+00   0.342 0.732631    
## GarageCars      1.515e+04  2.935e+03   5.162 2.79e-07 ***
## GarageCond_q    7.608e+02  4.137e+03   0.184 0.854106    
## GarageYrBlt    -1.528e+01  6.655e+00  -2.296 0.021813 *  
## GrLivArea       4.411e+01  4.914e+00   8.976  < 2e-16 ***
## HalfBath       -4.686e+02  2.627e+03  -0.178 0.858443    
## HeatingQC_q     1.357e+03  1.199e+03   1.132 0.257952    
## KitchenAbvGr   -1.549e+04  5.176e+03  -2.993 0.002807 ** 
## KitchenQual_q   1.341e+04  2.085e+03   6.434 1.70e-10 ***
## LotArea         4.128e-01  9.852e-02   4.190 2.96e-05 ***
## LotFrontage     9.013e+00  2.807e+01   0.321 0.748199    
## LowQualFinSF   -2.675e+01  1.951e+01  -1.371 0.170611    
## MasVnrArea      2.925e+01  5.819e+00   5.027 5.63e-07 ***
## MiscVal         2.387e-01  1.817e+00   0.131 0.895508    
## MoSold         -7.896e+01  3.365e+02  -0.235 0.814482    
## MSSubClass     -1.573e+02  2.629e+01  -5.985 2.74e-09 ***
## OpenPorchSF    -9.371e+00  1.482e+01  -0.632 0.527361    
## OverallCond     5.113e+03  1.038e+03   4.927 9.34e-07 ***
## OverallQual     1.520e+04  1.199e+03  12.682  < 2e-16 ***
## PoolArea       -2.954e+01  2.355e+01  -1.254 0.209908    
## ScreenPorch     5.503e+01  1.678e+01   3.280 0.001063 ** 
## TotalBsmtSF            NA         NA      NA       NA    
## TotRmsAbvGrd    4.565e+03  1.211e+03   3.769 0.000171 ***
## WoodDeckSF      2.651e+01  7.824e+00   3.388 0.000723 ***
## X1stFlrSF      -1.228e+00  5.306e+00  -0.231 0.817048    
## X2ndFlrSF              NA         NA      NA       NA    
## X3SsnPorch      2.230e+01  3.065e+01   0.728 0.466929    
## YearBuilt       2.432e+02  6.659e+01   3.653 0.000269 ***
## YearRemodAdd   -2.487e+01  7.007e+01  -0.355 0.722728    
## YrSold         -6.977e+02  6.862e+02  -1.017 0.309402    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 33860 on 1418 degrees of freedom
## Multiple R-squared:  0.8234, Adjusted R-squared:  0.8183 
## F-statistic: 161.3 on 41 and 1418 DF,  p-value: < 2.2e-16
###############################################

# Removing "TotalBsmtSF" and "X2ndFlrSF" from  "p2_train_vars"

p2_train_vars_2 <- subset(p2_train_vars, select = -c(TotalBsmtSF,X2ndFlrSF))

p2_train_regr_v2 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_2)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v2
## SalePrice ~ BedroomAbvGr + BsmtFinSF1 + BsmtFinSF2 + BsmtFinType2_q + 
##     BsmtFullBath + BsmtHalfBath + BsmtUnfSF + Electrical_q + 
##     EnclosedPorch + Fence_q + Fireplaces + Foundation_q + FullBath + 
##     GarageArea + GarageCars + GarageCond_q + GarageYrBlt + GrLivArea + 
##     HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + 
##     LotFrontage + LowQualFinSF + MasVnrArea + MiscVal + MoSold + 
##     MSSubClass + OpenPorchSF + OverallCond + OverallQual + PoolArea + 
##     ScreenPorch + TotRmsAbvGrd + WoodDeckSF + X1stFlrSF + X3SsnPorch + 
##     YearBuilt + YearRemodAdd + YrSold
#--------------- Linear Model Version 2 --------------

lm_2.lm <- lm((p2_train_regr_v2),data = p2_train)

summary(lm_2.lm)
## 
## Call:
## lm(formula = (p2_train_regr_v2), data = p2_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -472691  -16414   -1979   13200  297806 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     9.261e+05  1.385e+06   0.669 0.503705    
## BedroomAbvGr   -8.213e+03  1.683e+03  -4.881 1.17e-06 ***
## BsmtFinSF1      1.986e+01  4.669e+00   4.255 2.23e-05 ***
## BsmtFinSF2      1.577e+01  1.048e+01   1.505 0.132474    
## BsmtFinType2_q -7.895e+02  1.703e+03  -0.464 0.642987    
## BsmtFullBath    8.031e+03  2.567e+03   3.129 0.001792 ** 
## BsmtHalfBath    1.919e+03  3.998e+03   0.480 0.631350    
## BsmtUnfSF       9.120e+00  4.279e+00   2.131 0.033221 *  
## Electrical_q   -3.220e+03  2.401e+03  -1.341 0.180057    
## EnclosedPorch   2.458e+00  1.653e+01   0.149 0.881788    
## Fence_q        -1.052e+03  7.984e+02  -1.317 0.187955    
## Fireplaces      4.547e+03  1.734e+03   2.622 0.008841 ** 
## Foundation_q   -3.347e+03  1.714e+03  -1.953 0.051019 .  
## FullBath        3.184e+03  2.762e+03   1.153 0.249170    
## GarageArea      3.283e+00  9.608e+00   0.342 0.732631    
## GarageCars      1.515e+04  2.935e+03   5.162 2.79e-07 ***
## GarageCond_q    7.608e+02  4.137e+03   0.184 0.854106    
## GarageYrBlt    -1.528e+01  6.655e+00  -2.296 0.021813 *  
## GrLivArea       4.411e+01  4.914e+00   8.976  < 2e-16 ***
## HalfBath       -4.686e+02  2.627e+03  -0.178 0.858443    
## HeatingQC_q     1.357e+03  1.199e+03   1.132 0.257952    
## KitchenAbvGr   -1.549e+04  5.176e+03  -2.993 0.002807 ** 
## KitchenQual_q   1.341e+04  2.085e+03   6.434 1.70e-10 ***
## LotArea         4.128e-01  9.852e-02   4.190 2.96e-05 ***
## LotFrontage     9.013e+00  2.807e+01   0.321 0.748199    
## LowQualFinSF   -2.675e+01  1.951e+01  -1.371 0.170611    
## MasVnrArea      2.925e+01  5.819e+00   5.027 5.63e-07 ***
## MiscVal         2.387e-01  1.817e+00   0.131 0.895508    
## MoSold         -7.896e+01  3.365e+02  -0.235 0.814482    
## MSSubClass     -1.573e+02  2.629e+01  -5.985 2.74e-09 ***
## OpenPorchSF    -9.371e+00  1.482e+01  -0.632 0.527361    
## OverallCond     5.113e+03  1.038e+03   4.927 9.34e-07 ***
## OverallQual     1.520e+04  1.199e+03  12.682  < 2e-16 ***
## PoolArea       -2.954e+01  2.355e+01  -1.254 0.209908    
## ScreenPorch     5.503e+01  1.678e+01   3.280 0.001063 ** 
## TotRmsAbvGrd    4.565e+03  1.211e+03   3.769 0.000171 ***
## WoodDeckSF      2.651e+01  7.824e+00   3.388 0.000723 ***
## X1stFlrSF      -1.228e+00  5.306e+00  -0.231 0.817048    
## X3SsnPorch      2.230e+01  3.065e+01   0.728 0.466929    
## YearBuilt       2.432e+02  6.659e+01   3.653 0.000269 ***
## YearRemodAdd   -2.487e+01  7.007e+01  -0.355 0.722728    
## YrSold         -6.977e+02  6.862e+02  -1.017 0.309402    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 33860 on 1418 degrees of freedom
## Multiple R-squared:  0.8234, Adjusted R-squared:  0.8183 
## F-statistic: 161.3 on 41 and 1418 DF,  p-value: < 2.2e-16
###############################################

Testing for and Freeing From Multicollinearity among Variables

Multicollinearity occurs when two or more predictor variables are highly correlated to each other, such that they do not provide unique or independent information in the regression model.

If the degree of correlation is high enough between variables, it can cause problems when fitting and interpreting the regression model.

To test this model for Multicollinearity we will employ the “imcdiag” function from the “mctest” library and examine the Variance Inflation Factor (VIF) score.

Note : Scores over 5 are moderately multicollinear. Scores over 10 are very problematic

using the VIF measure we see that most of the predictor variables posses low VIF scores indicating that they are not very correlated, but the following variables are moderately to problematic :

GarageCond_q - VIF score is 11.3 —- Problematic —- Will be removed from the model

GarageYrBlt - VIF score is 11.60 —- Problematic —- Will be removed from the model

GrLivArea- VIF score is 8.48 —- moderately multicollinear —-

imcdiag(lm_2.lm)
## 
## Call:
## imcdiag(mod = lm_2.lm)
## 
## 
## All Individual Multicollinearity Diagnostics Result
## 
##                    VIF    TOL       Wi       Fi Leamer    CVIF Klein   IND1
## BedroomAbvGr    2.3969 0.4172  49.5562  50.8626 0.6459 -0.0923     0 0.0118
## BsmtFinSF1      5.7687 0.1733 169.1698 173.6298 0.4164 -0.2220     1 0.0049
## BsmtFinSF2      3.6362 0.2750  93.5204  95.9860 0.5244 -0.1400     0 0.0078
## BsmtFinType2_q  2.9377 0.3404  68.7416  70.5539 0.5834 -0.1131     0 0.0096
## BsmtFullBath    2.2574 0.4430  44.6072  45.7832 0.6656 -0.0869     0 0.0125
## BsmtHalfBath    1.1592 0.8627   5.6461   5.7949 0.9288 -0.0446     0 0.0243
## BsmtUnfSF       4.5482 0.2199 125.8720 129.1905 0.4689 -0.1751     0 0.0062
## Electrical_q    1.2624 0.7922   9.3080   9.5534 0.8900 -0.0486     0 0.0223
## EnclosedPorch   1.2981 0.7704  10.5746  10.8533 0.8777 -0.0500     0 0.0217
## Fence_q         1.1765 0.8500   6.2612   6.4262 0.9219 -0.0453     0 0.0240
## Fireplaces      1.5906 0.6287  20.9503  21.5027 0.7929 -0.0612     0 0.0177
## Foundation_q    1.9496 0.5129  33.6875  34.5756 0.7162 -0.0750     0 0.0145
## FullBath        2.9457 0.3395  69.0246  70.8443 0.5826 -0.1134     0 0.0096
## GarageArea      5.3691 0.1863 154.9947 159.0809 0.4316 -0.2066     0 0.0053
## GarageCars      6.1208 0.1634 181.6593 186.4486 0.4042 -0.2356     1 0.0046
## GarageCond_q   11.2780 0.0887 364.6126 374.2252 0.2978 -0.4341     1 0.0025
## GarageYrBlt    11.5973 0.0862 375.9403 385.8515 0.2936 -0.4464     1 0.0024
## GrLivArea       8.4839 0.1179 265.4925 272.4919 0.3433 -0.3265     1 0.0033
## HalfBath        2.2207 0.4503  43.3030  44.4447 0.6711 -0.0855     0 0.0127
## HeatingQC_q     1.6831 0.5941  24.2324  24.8713 0.7708 -0.0648     0 0.0167
## KitchenAbvGr    1.6549 0.6043  23.2322  23.8446 0.7773 -0.0637     0 0.0170
## KitchenQual_q   2.4362 0.4105  50.9487  52.2919 0.6407 -0.0938     0 0.0116
## LotArea         1.2302 0.8128   8.1681   8.3834 0.9016 -0.0474     0 0.0229
## LotFrontage     1.2047 0.8301   7.2616   7.4530 0.9111 -0.0464     0 0.0234
## LowQualFinSF    1.1450 0.8734   5.1441   5.2797 0.9345 -0.0441     0 0.0246
## MasVnrArea      1.4070 0.7107  14.4380  14.8186 0.8431 -0.0542     0 0.0200
## MiscVal         1.0339 0.9672   1.2034   1.2351 0.9835 -0.0398     0 0.0273
## MoSold          1.0528 0.9498   1.8734   1.9228 0.9746 -0.0405     0 0.0268
## MSSubClass      1.5729 0.6358  20.3231  20.8589 0.7974 -0.0605     0 0.0179
## OpenPorchSF     1.2271 0.8149   8.0573   8.2697 0.9027 -0.0472     0 0.0230
## OverallCond     1.6967 0.5894  24.7163  25.3679 0.7677 -0.0653     0 0.0166
## OverallQual     3.4975 0.2859  88.5980  90.9338 0.5347 -0.1346     0 0.0081
## PoolArea        1.1386 0.8783   4.9172   5.0468 0.9372 -0.0438     0 0.0248
## ScreenPorch     1.1134 0.8981   4.0238   4.1299 0.9477 -0.0429     0 0.0253
## TotRmsAbvGrd    4.9303 0.2028 139.4256 143.1014 0.4504 -0.1898     0 0.0057
## WoodDeckSF      1.2235 0.8173   7.9287   8.1377 0.9041 -0.0471     0 0.0230
## X1stFlrSF       5.3540 0.1868 154.4569 158.5290 0.4322 -0.2061     0 0.0053
## X3SsnPorch      1.0272 0.9735   0.9661   0.9915 0.9867 -0.0395     0 0.0274
## YearBuilt       5.1461 0.1943 147.0833 150.9610 0.4408 -0.1981     0 0.0055
## YearRemodAdd    2.6625 0.3756  58.9784  60.5332 0.6128 -0.1025     0 0.0106
## YrSold          1.0567 0.9464   2.0098   2.0628 0.9728 -0.0407     0 0.0267
##                  IND2
## BedroomAbvGr   1.2757
## BsmtFinSF1     1.8095
## BsmtFinSF2     1.5870
## BsmtFinType2_q 1.4439
## BsmtFullBath   1.2193
## BsmtHalfBath   0.3006
## BsmtUnfSF      1.7077
## Electrical_q   0.4550
## EnclosedPorch  0.5027
## Fence_q        0.3284
## Fireplaces     0.8128
## Foundation_q   1.0662
## FullBath       1.4459
## GarageArea     1.7813
## GarageCars     1.8314
## GarageCond_q   1.9949
## GarageYrBlt    2.0002
## GrLivArea      1.9310
## HalfBath       1.2032
## HeatingQC_q    0.8884
## KitchenAbvGr   0.8662
## KitchenQual_q  1.2905
## LotArea        0.4097
## LotFrontage    0.3719
## LowQualFinSF   0.2772
## MasVnrArea     0.6332
## MiscVal        0.0718
## MoSold         0.1098
## MSSubClass     0.7973
## OpenPorchSF    0.4052
## OverallCond    0.8989
## OverallQual    1.5631
## PoolArea       0.2665
## ScreenPorch    0.2230
## TotRmsAbvGrd   1.7450
## WoodDeckSF     0.3999
## X1stFlrSF      1.7801
## X3SsnPorch     0.0580
## YearBuilt      1.7636
## YearRemodAdd   1.3668
## YrSold         0.1174
## 
## 1 --> COLLINEARITY is detected by the test 
## 0 --> COLLINEARITY is not detected by the test
## 
## BsmtFinSF2 , BsmtFinType2_q , BsmtHalfBath , Electrical_q , EnclosedPorch , Fence_q , Foundation_q , FullBath , GarageArea , GarageCond_q , HalfBath , HeatingQC_q , LotFrontage , LowQualFinSF , MiscVal , MoSold , OpenPorchSF , PoolArea , X1stFlrSF , X3SsnPorch , YearRemodAdd , YrSold , coefficient(s) are non-significant may be due to multicollinearity
## 
## R-square of y on all x: 0.8234 
## 
## * use method argument to check which regressors may be the reason of collinearity
## ===================================

Regression Modeling V3

# We will remove the "GarageCond_q" and "GarageYrBlt" variables and recompute the Linear Model

p2_train_vars_3 <- subset(p2_train_vars_2, select = -c(GarageCond_q,GarageYrBlt))

p2_train_regr_v3 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_3)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v3
## SalePrice ~ BedroomAbvGr + BsmtFinSF1 + BsmtFinSF2 + BsmtFinType2_q + 
##     BsmtFullBath + BsmtHalfBath + BsmtUnfSF + Electrical_q + 
##     EnclosedPorch + Fence_q + Fireplaces + Foundation_q + FullBath + 
##     GarageArea + GarageCars + GrLivArea + HalfBath + HeatingQC_q + 
##     KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + 
##     MasVnrArea + MiscVal + MoSold + MSSubClass + OpenPorchSF + 
##     OverallCond + OverallQual + PoolArea + ScreenPorch + TotRmsAbvGrd + 
##     WoodDeckSF + X1stFlrSF + X3SsnPorch + YearBuilt + YearRemodAdd + 
##     YrSold
#--------------- Linear Model Version 3 --------------

lm_3.lm <- lm((p2_train_regr_v3),data = p2_train)

summary(lm_3.lm)
## 
## Call:
## lm(formula = (p2_train_regr_v3), data = p2_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -478319  -16598   -1832   13242  301926 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     1.019e+06  1.397e+06   0.729 0.465881    
## BedroomAbvGr   -8.319e+03  1.696e+03  -4.904 1.05e-06 ***
## BsmtFinSF1      1.991e+01  4.706e+00   4.231 2.47e-05 ***
## BsmtFinSF2      1.308e+01  1.056e+01   1.239 0.215636    
## BsmtFinType2_q -4.872e+02  1.718e+03  -0.284 0.776729    
## BsmtFullBath    8.926e+03  2.585e+03   3.453 0.000571 ***
## BsmtHalfBath    2.044e+03  4.035e+03   0.507 0.612552    
## BsmtUnfSF       9.721e+00  4.301e+00   2.260 0.023979 *  
## Electrical_q   -3.068e+03  2.399e+03  -1.279 0.201193    
## EnclosedPorch   1.573e+00  1.668e+01   0.094 0.924866    
## Fence_q        -1.296e+03  8.045e+02  -1.611 0.107360    
## Fireplaces      4.138e+03  1.748e+03   2.368 0.018012 *  
## Foundation_q   -3.794e+03  1.727e+03  -2.197 0.028199 *  
## FullBath        4.213e+03  2.781e+03   1.515 0.130000    
## GarageArea     -3.242e+00  9.580e+00  -0.338 0.735097    
## GarageCars      1.026e+04  2.808e+03   3.654 0.000268 ***
## GrLivArea       4.378e+01  4.957e+00   8.831  < 2e-16 ***
## HalfBath       -1.437e+02  2.650e+03  -0.054 0.956770    
## HeatingQC_q     1.149e+03  1.208e+03   0.951 0.341950    
## KitchenAbvGr   -1.289e+04  5.198e+03  -2.479 0.013276 *  
## KitchenQual_q   1.348e+04  2.103e+03   6.412 1.95e-10 ***
## LotArea         4.195e-01  9.943e-02   4.219 2.61e-05 ***
## LotFrontage     1.968e+01  2.826e+01   0.696 0.486339    
## LowQualFinSF   -1.555e+01  1.954e+01  -0.796 0.426233    
## MasVnrArea      3.216e+01  5.847e+00   5.500 4.49e-08 ***
## MiscVal         9.816e-02  1.834e+00   0.054 0.957319    
## MoSold         -8.641e+01  3.396e+02  -0.254 0.799172    
## MSSubClass     -1.567e+02  2.653e+01  -5.908 4.33e-09 ***
## OpenPorchSF    -4.484e+00  1.493e+01  -0.300 0.764012    
## OverallCond     4.533e+03  1.036e+03   4.376 1.30e-05 ***
## OverallQual     1.521e+04  1.210e+03  12.573  < 2e-16 ***
## PoolArea       -3.086e+01  2.373e+01  -1.301 0.193611    
## ScreenPorch     5.328e+01  1.692e+01   3.148 0.001676 ** 
## TotRmsAbvGrd    4.648e+03  1.222e+03   3.804 0.000149 ***
## WoodDeckSF      2.701e+01  7.892e+00   3.422 0.000639 ***
## X1stFlrSF      -7.930e-02  5.337e+00  -0.015 0.988148    
## X3SsnPorch      1.972e+01  3.093e+01   0.638 0.523896    
## YearBuilt       2.204e+02  6.665e+01   3.307 0.000967 ***
## YearRemodAdd    1.820e+01  6.994e+01   0.260 0.794684    
## YrSold         -7.728e+02  6.925e+02  -1.116 0.264615    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 34180 on 1420 degrees of freedom
## Multiple R-squared:  0.8198, Adjusted R-squared:  0.8149 
## F-statistic: 165.7 on 39 and 1420 DF,  p-value: < 2.2e-16
###############################################

imcdiag(lm_3.lm)
## 
## Call:
## imcdiag(mod = lm_3.lm)
## 
## 
## All Individual Multicollinearity Diagnostics Result
## 
##                   VIF    TOL       Wi       Fi Leamer    CVIF Klein   IND1
## BedroomAbvGr   2.3914 0.4182  52.0308  53.4747 0.6467 -0.0968     0 0.0112
## BsmtFinSF1     5.7536 0.1738 177.7583 182.6911 0.4169 -0.2329     1 0.0046
## BsmtFinSF2     3.6237 0.2760  98.1124 100.8350 0.5253 -0.1467     0 0.0074
## BsmtFinType2_q 2.9336 0.3409  72.3071  74.3136 0.5838 -0.1188     0 0.0091
## BsmtFullBath   2.2470 0.4450  46.6298  47.9238 0.6671 -0.0910     0 0.0119
## BsmtHalfBath   1.1591 0.8627   5.9491   6.1142 0.9288 -0.0469     0 0.0231
## BsmtUnfSF      4.5116 0.2217 131.3137 134.9577 0.4708 -0.1826     0 0.0059
## Electrical_q   1.2370 0.8084   8.8621   9.1080 0.8991 -0.0501     0 0.0216
## EnclosedPorch  1.2977 0.7706  11.1336  11.4426 0.8778 -0.0525     0 0.0206
## Fence_q        1.1725 0.8529   6.4519   6.6309 0.9235 -0.0475     0 0.0228
## Fireplaces     1.5850 0.6309  21.8767  22.4838 0.7943 -0.0642     0 0.0169
## Foundation_q   1.9443 0.5143  35.3132  36.2932 0.7172 -0.0787     0 0.0138
## FullBath       2.9314 0.3411  72.2248  74.2290 0.5841 -0.1187     0 0.0091
## GarageArea     5.2399 0.1908 158.5491 162.9488 0.4369 -0.2121     0 0.0051
## GarageCars     5.5006 0.1818 168.2999 172.9702 0.4264 -0.2227     0 0.0049
## GrLivArea      8.4742 0.1180 279.4952 287.2511 0.3435 -0.3430     1 0.0032
## HalfBath       2.2176 0.4509  45.5307  46.7941 0.6715 -0.0898     0 0.0121
## HeatingQC_q    1.6788 0.5957  25.3823  26.0867 0.7718 -0.0680     0 0.0159
## KitchenAbvGr   1.6384 0.6104  23.8729  24.5353 0.7812 -0.0663     0 0.0163
## KitchenQual_q  2.4328 0.4110  53.5805  55.0673 0.6411 -0.0985     0 0.0110
## LotArea        1.2300 0.8130   8.6024   8.8411 0.9017 -0.0498     0 0.0217
## LotFrontage    1.1985 0.8344   7.4214   7.6274 0.9135 -0.0485     0 0.0223
## LowQualFinSF   1.1276 0.8868   4.7711   4.9035 0.9417 -0.0456     0 0.0237
## MasVnrArea     1.3945 0.7171  14.7538  15.1633 0.8468 -0.0565     0 0.0192
## MiscVal        1.0337 0.9674   1.2593   1.2943 0.9836 -0.0418     0 0.0259
## MoSold         1.0527 0.9500   1.9695   2.0242 0.9747 -0.0426     0 0.0254
## MSSubClass     1.5726 0.6359  21.4106  22.0048 0.7974 -0.0637     0 0.0170
## OpenPorchSF    1.2224 0.8180   8.3182   8.5490 0.9045 -0.0495     0 0.0219
## OverallCond    1.6596 0.6026  24.6647  25.3491 0.7762 -0.0672     0 0.0161
## OverallQual    3.4964 0.2860  93.3534  95.9439 0.5348 -0.1415     0 0.0076
## PoolArea       1.1352 0.8809   5.0546   5.1949 0.9386 -0.0460     0 0.0236
## ScreenPorch    1.1118 0.8994   4.1807   4.2967 0.9484 -0.0450     0 0.0241
## TotRmsAbvGrd   4.9256 0.2030 146.7952 150.8688 0.4506 -0.1994     0 0.0054
## WoodDeckSF     1.2221 0.8183   8.3050   8.5354 0.9046 -0.0495     0 0.0219
## X1stFlrSF      5.3168 0.1881 161.4269 165.9064 0.4337 -0.2152     0 0.0050
## X3SsnPorch     1.0268 0.9739   1.0040   1.0319 0.9868 -0.0416     0 0.0260
## YearBuilt      5.0606 0.1976 151.8454 156.0591 0.4445 -0.2049     0 0.0053
## YearRemodAdd   2.6038 0.3841  59.9727  61.6369 0.6197 -0.1054     0 0.0103
## YrSold         1.0562 0.9468   2.1017   2.1600 0.9730 -0.0428     0 0.0253
##                  IND2
## BedroomAbvGr   1.3522
## BsmtFinSF1     1.9201
## BsmtFinSF2     1.6826
## BsmtFinType2_q 1.5318
## BsmtFullBath   1.2897
## BsmtHalfBath   0.3190
## BsmtUnfSF      1.8089
## Electrical_q   0.4452
## EnclosedPorch  0.5332
## Fence_q        0.3420
## Fireplaces     0.8578
## Foundation_q   1.1287
## FullBath       1.5312
## GarageArea     1.8805
## GarageCars     1.9015
## GrLivArea      2.0497
## HalfBath       1.2760
## HeatingQC_q    0.9396
## KitchenAbvGr   0.9055
## KitchenQual_q  1.3687
## LotArea        0.4346
## LotFrontage    0.3848
## LowQualFinSF   0.2630
## MasVnrArea     0.6575
## MiscVal        0.0757
## MoSold         0.1163
## MSSubClass     0.8461
## OpenPorchSF    0.4229
## OverallCond    0.9236
## OverallQual    1.6593
## PoolArea       0.2767
## ScreenPorch    0.2337
## TotRmsAbvGrd   1.8522
## WoodDeckSF     0.4223
## X1stFlrSF      1.8869
## X3SsnPorch     0.0608
## YearBuilt      1.8647
## YearRemodAdd   1.4314
## YrSold         0.1237
## 
## 1 --> COLLINEARITY is detected by the test 
## 0 --> COLLINEARITY is not detected by the test
## 
## BsmtFinSF2 , BsmtFinType2_q , BsmtHalfBath , Electrical_q , EnclosedPorch , Fence_q , FullBath , GarageArea , HalfBath , HeatingQC_q , LotFrontage , LowQualFinSF , MiscVal , MoSold , OpenPorchSF , PoolArea , X1stFlrSF , X3SsnPorch , YearRemodAdd , YrSold , coefficient(s) are non-significant may be due to multicollinearity
## 
## R-square of y on all x: 0.8198 
## 
## * use method argument to check which regressors may be the reason of collinearity
## ===================================

Version 3 Discussion :

Note that after computing Linear Model V3, we have Multiple - \(R^{2} = 0.8198\) and \(R^{2} = 0.8149\) - Not a significant difference from V2 modeling

Also - The Multicollinearity test shows the VIF scores for the following variables to be > 8

“GrLivArea”

These will be removed in V4 :


Regression Modeling V4

# We will remove the "GrLivArea" variables : 

p2_train_vars_4 <- subset(p2_train_vars_3, select = -c(GrLivArea))

p2_train_regr_v4 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_4)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v4
## SalePrice ~ BedroomAbvGr + BsmtFinSF1 + BsmtFinSF2 + BsmtFinType2_q + 
##     BsmtFullBath + BsmtHalfBath + BsmtUnfSF + Electrical_q + 
##     EnclosedPorch + Fence_q + Fireplaces + Foundation_q + FullBath + 
##     GarageArea + GarageCars + HalfBath + HeatingQC_q + KitchenAbvGr + 
##     KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + MasVnrArea + 
##     MiscVal + MoSold + MSSubClass + OpenPorchSF + OverallCond + 
##     OverallQual + PoolArea + ScreenPorch + TotRmsAbvGrd + WoodDeckSF + 
##     X1stFlrSF + X3SsnPorch + YearBuilt + YearRemodAdd + YrSold
#--------------- Linear Model Version 4 --------------

lm_4.lm <- lm((p2_train_regr_v4),data = p2_train)

summary(lm_4.lm)
## 
## Call:
## lm(formula = (p2_train_regr_v4), data = p2_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -446391  -17796   -2727   14175  341600 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     1.245e+06  1.435e+06   0.868 0.385580    
## BedroomAbvGr   -6.221e+03  1.724e+03  -3.607 0.000320 ***
## BsmtFinSF1      2.145e+01  4.829e+00   4.443 9.55e-06 ***
## BsmtFinSF2      1.211e+01  1.084e+01   1.117 0.264319    
## BsmtFinType2_q -1.880e+02  1.763e+03  -0.107 0.915102    
## BsmtFullBath    7.746e+03  2.650e+03   2.923 0.003526 ** 
## BsmtHalfBath    9.753e+02  4.141e+03   0.236 0.813843    
## BsmtUnfSF       8.507e+00  4.414e+00   1.927 0.054149 .  
## Electrical_q   -3.719e+03  2.462e+03  -1.511 0.131080    
## EnclosedPorch   9.446e+00  1.710e+01   0.552 0.580759    
## Fence_q        -1.432e+03  8.258e+02  -1.734 0.083097 .  
## Fireplaces      5.829e+03  1.783e+03   3.268 0.001109 ** 
## Foundation_q   -5.692e+03  1.760e+03  -3.235 0.001245 ** 
## FullBath        1.332e+04  2.652e+03   5.022 5.76e-07 ***
## GarageArea      6.983e+00  9.765e+00   0.715 0.474665    
## GarageCars      8.707e+03  2.878e+03   3.025 0.002527 ** 
## HalfBath        1.119e+04  2.380e+03   4.703 2.81e-06 ***
## HeatingQC_q     2.142e+03  1.235e+03   1.734 0.083201 .  
## KitchenAbvGr   -1.935e+04  5.284e+03  -3.662 0.000260 ***
## KitchenQual_q   1.412e+04  2.158e+03   6.543 8.37e-11 ***
## LotArea         4.944e-01  1.017e-01   4.860 1.30e-06 ***
## LotFrontage     2.581e+01  2.901e+01   0.890 0.373782    
## LowQualFinSF    1.069e+01  1.983e+01   0.539 0.590117    
## MasVnrArea      3.772e+01  5.968e+00   6.320 3.49e-10 ***
## MiscVal         6.330e-02  1.883e+00   0.034 0.973184    
## MoSold          1.115e+01  3.485e+02   0.032 0.974481    
## MSSubClass     -1.064e+02  2.660e+01  -4.001 6.62e-05 ***
## OpenPorchSF     6.207e+00  1.528e+01   0.406 0.684645    
## OverallCond     4.143e+03  1.063e+03   3.899 0.000101 ***
## OverallQual     1.695e+04  1.226e+03  13.830  < 2e-16 ***
## PoolArea       -1.204e+00  2.412e+01  -0.050 0.960196    
## ScreenPorch     5.400e+01  1.737e+01   3.108 0.001921 ** 
## TotRmsAbvGrd    9.574e+03  1.116e+03   8.578  < 2e-16 ***
## WoodDeckSF      3.161e+01  8.086e+00   3.910 9.68e-05 ***
## X1stFlrSF       1.735e+01  5.092e+00   3.407 0.000675 ***
## X3SsnPorch      2.129e+01  3.176e+01   0.670 0.502755    
## YearBuilt       2.657e+01  6.461e+01   0.411 0.680973    
## YearRemodAdd    2.468e+01  7.181e+01   0.344 0.731161    
## YrSold         -7.093e+02  7.109e+02  -0.998 0.318553    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 35090 on 1421 degrees of freedom
## Multiple R-squared:  0.8099, Adjusted R-squared:  0.8049 
## F-statistic: 159.4 on 38 and 1421 DF,  p-value: < 2.2e-16
imcdiag(lm_4.lm)
## 
## Call:
## imcdiag(mod = lm_4.lm)
## 
## 
## All Individual Multicollinearity Diagnostics Result
## 
##                   VIF    TOL       Wi       Fi Leamer    CVIF Klein   IND1
## BedroomAbvGr   2.3445 0.4265  51.6719  53.1446 0.6531 -0.1128     0 0.0111
## BsmtFinSF1     5.7457 0.1740 182.3872 187.5853 0.4172 -0.2766     1 0.0045
## BsmtFinSF2     3.6233 0.2760 100.8198 103.6932 0.5253 -0.1744     0 0.0072
## BsmtFinType2_q 2.9325 0.3410  74.2697  76.3864 0.5840 -0.1412     0 0.0089
## BsmtFullBath   2.2410 0.4462  47.6933  49.0526 0.6680 -0.1079     0 0.0116
## BsmtHalfBath   1.1580 0.8635   6.0741   6.2472 0.9293 -0.0557     0 0.0225
## BsmtUnfSF      4.5070 0.2219 134.7807 138.6220 0.4710 -0.2169     0 0.0058
## Electrical_q   1.2358 0.8092   9.0630   9.3213 0.8995 -0.0595     0 0.0211
## EnclosedPorch  1.2940 0.7728  11.3001  11.6222 0.8791 -0.0623     0 0.0201
## Fence_q        1.1721 0.8532   6.6144   6.8029 0.9237 -0.0564     0 0.0222
## Fireplaces     1.5660 0.6386  21.7532  22.3732 0.7991 -0.0754     0 0.0166
## Foundation_q   1.9142 0.5224  35.1365  36.1379 0.7228 -0.0921     0 0.0136
## FullBath       2.5286 0.3955  58.7463  60.4206 0.6289 -0.1217     0 0.0103
## GarageArea     5.1634 0.1937 160.0077 164.5680 0.4401 -0.2485     0 0.0050
## GarageCars     5.4790 0.1825 172.1398 177.0459 0.4272 -0.2637     1 0.0047
## HalfBath       1.6971 0.5892  26.7905  27.5540 0.7676 -0.0817     0 0.0153
## HeatingQC_q    1.6642 0.6009  25.5282  26.2558 0.7752 -0.0801     0 0.0156
## KitchenAbvGr   1.6059 0.6227  23.2880  23.9517 0.7891 -0.0773     0 0.0162
## KitchenQual_q  2.4300 0.4115  54.9577  56.5240 0.6415 -0.1170     0 0.0107
## LotArea        1.2211 0.8189   8.4978   8.7400 0.9049 -0.0588     0 0.0213
## LotFrontage    1.1977 0.8349   7.5996   7.8162 0.9137 -0.0577     0 0.0217
## LowQualFinSF   1.1015 0.9078   3.9018   4.0130 0.9528 -0.0530     0 0.0236
## MasVnrArea     1.3784 0.7255  14.5417  14.9561 0.8518 -0.0663     0 0.0189
## MiscVal        1.0337 0.9674   1.2941   1.3310 0.9836 -0.0498     0 0.0252
## MoSold         1.0516 0.9510   1.9813   2.0378 0.9752 -0.0506     0 0.0247
## MSSubClass     1.5001 0.6666  19.2206  19.7684 0.8165 -0.0722     0 0.0173
## OpenPorchSF    1.2144 0.8234   8.2402   8.4751 0.9074 -0.0585     0 0.0214
## OverallCond    1.6566 0.6037  25.2333  25.9525 0.7770 -0.0797     0 0.0157
## OverallQual    3.4038 0.2938  92.3855  95.0185 0.5420 -0.1638     0 0.0076
## PoolArea       1.1124 0.8989   4.3211   4.4442 0.9481 -0.0535     0 0.0234
## ScreenPorch    1.1118 0.8995   4.2957   4.4181 0.9484 -0.0535     0 0.0234
## TotRmsAbvGrd   3.8987 0.2565 111.4045 114.5796 0.5065 -0.1877     0 0.0067
## WoodDeckSF     1.2168 0.8219   8.3307   8.5681 0.9066 -0.0586     0 0.0214
## X1stFlrSF      4.5901 0.2179 137.9758 141.9082 0.4668 -0.2209     0 0.0057
## X3SsnPorch     1.0268 0.9739   1.0306   1.0600 0.9869 -0.0494     0 0.0253
## YearBuilt      4.5118 0.2216 134.9689 138.8155 0.4708 -0.2172     0 0.0058
## YearRemodAdd   2.6035 0.3841  61.6259  63.3823 0.6198 -0.1253     0 0.0100
## YrSold         1.0561 0.9469   2.1556   2.2171 0.9731 -0.0508     0 0.0246
##                  IND2
## BedroomAbvGr   1.4110
## BsmtFinSF1     2.0322
## BsmtFinSF2     1.7814
## BsmtFinType2_q 1.6214
## BsmtFullBath   1.3625
## BsmtHalfBath   0.3358
## BsmtUnfSF      1.9145
## Electrical_q   0.4695
## EnclosedPorch  0.5591
## Fence_q        0.3613
## Fireplaces     0.8893
## Foundation_q   1.1751
## FullBath       1.4874
## GarageArea     1.9839
## GarageCars     2.0114
## HalfBath       1.0106
## HeatingQC_q    0.9820
## KitchenAbvGr   0.9284
## KitchenQual_q  1.4479
## LotArea        0.4455
## LotFrontage    0.4062
## LowQualFinSF   0.2268
## MasVnrArea     0.6754
## MiscVal        0.0801
## MoSold         0.1206
## MSSubClass     0.8203
## OpenPorchSF    0.4344
## OverallCond    0.9752
## OverallQual    1.7376
## PoolArea       0.2487
## ScreenPorch    0.2474
## TotRmsAbvGrd   1.8293
## WoodDeckSF     0.4383
## X1stFlrSF      1.9244
## X3SsnPorch     0.0643
## YearBuilt      1.9151
## YearRemodAdd   1.5154
## YrSold         0.1307
## 
## 1 --> COLLINEARITY is detected by the test 
## 0 --> COLLINEARITY is not detected by the test
## 
## BsmtFinSF2 , BsmtFinType2_q , BsmtHalfBath , BsmtUnfSF , Electrical_q , EnclosedPorch , Fence_q , GarageArea , HeatingQC_q , LotFrontage , LowQualFinSF , MiscVal , MoSold , OpenPorchSF , PoolArea , X3SsnPorch , YearBuilt , YearRemodAdd , YrSold , coefficient(s) are non-significant may be due to multicollinearity
## 
## R-square of y on all x: 0.8099 
## 
## * use method argument to check which regressors may be the reason of collinearity
## ===================================

Version 4 Discussion :

Note that after computing Linear Model V4, we have Multiple - \(R^{2} = 0.8099\) and \(R^{2} = 0.8049\) -

Note - As a final tuning to the model, we will remove the variables with VIF Scores > 3 in V5 of the Model. - Taking a more Conservative Approach as suggested by some researchers - https://quantifyinghealth.com/vif-threshold/


Regression Modeling V5

# We will remove the "BsmtFinSF1", "BsmtFinSF2, "BsmtUnfSF", "GarageArea", "GarageCars", "OverallQual", "TotRmsAbvGrd", "X1stFlrSF", "YearBuilt" variables : 

p2_train_vars_5 <- subset(p2_train_vars_4, select = -c(BsmtFinSF1,BsmtFinSF2,BsmtUnfSF,GarageArea,GarageCars,OverallQual,TotRmsAbvGrd,X1stFlrSF,YearBuilt))

p2_train_regr_v5 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_5)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v5
## SalePrice ~ BedroomAbvGr + BsmtFinType2_q + BsmtFullBath + BsmtHalfBath + 
##     Electrical_q + EnclosedPorch + Fence_q + Fireplaces + Foundation_q + 
##     FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + 
##     LotArea + LotFrontage + LowQualFinSF + MasVnrArea + MiscVal + 
##     MoSold + MSSubClass + OpenPorchSF + OverallCond + PoolArea + 
##     ScreenPorch + WoodDeckSF + X3SsnPorch + YearRemodAdd + YrSold
#--------------- Linear Model Version 5 --------------

lm_5.lm <- lm((p2_train_regr_v5),data = p2_train)

summary(lm_5.lm)
## 
## Call:
## lm(formula = (p2_train_regr_v5), data = p2_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -348505  -24685   -3607   19110  381768 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     2.594e+06  1.729e+06   1.501 0.133664    
## BedroomAbvGr    1.727e+03  1.671e+03   1.034 0.301320    
## BsmtFinType2_q  2.316e+01  1.315e+03   0.018 0.985952    
## BsmtFullBath    1.896e+04  2.404e+03   7.888 6.06e-15 ***
## BsmtHalfBath    2.940e+03  4.859e+03   0.605 0.545166    
## Electrical_q   -9.376e+02  2.922e+03  -0.321 0.748349    
## EnclosedPorch   1.846e+01  1.930e+01   0.957 0.338954    
## Fence_q        -2.196e+03  9.962e+02  -2.205 0.027631 *  
## Fireplaces      2.023e+04  1.998e+03  10.127  < 2e-16 ***
## Foundation_q   -8.058e+03  1.933e+03  -4.169 3.25e-05 ***
## FullBath        3.456e+04  2.956e+03  11.692  < 2e-16 ***
## HalfBath        1.574e+04  2.523e+03   6.237 5.85e-10 ***
## HeatingQC_q     3.496e+03  1.480e+03   2.362 0.018305 *  
## KitchenAbvGr   -9.404e+03  5.833e+03  -1.612 0.107172    
## KitchenQual_q   3.477e+04  2.417e+03  14.383  < 2e-16 ***
## LotArea         6.541e-01  1.216e-01   5.381 8.63e-08 ***
## LotFrontage     1.617e+02  3.446e+01   4.692 2.97e-06 ***
## LowQualFinSF    3.620e+01  2.343e+01   1.545 0.122576    
## MasVnrArea      8.466e+01  6.802e+00  12.446  < 2e-16 ***
## MiscVal         1.326e+00  2.276e+00   0.583 0.560242    
## MoSold          6.738e+01  4.213e+02   0.160 0.872944    
## MSSubClass     -1.824e+02  3.008e+01  -6.065 1.68e-09 ***
## OpenPorchSF     5.187e+01  1.817e+01   2.855 0.004372 ** 
## OverallCond     2.469e+03  1.142e+03   2.162 0.030755 *  
## PoolArea        2.082e+01  2.894e+01   0.720 0.471877    
## ScreenPorch     7.410e+01  2.090e+01   3.546 0.000403 ***
## WoodDeckSF      5.198e+01  9.735e+00   5.340 1.08e-07 ***
## X3SsnPorch      3.561e+01  3.842e+01   0.927 0.354057    
## YearRemodAdd    1.829e+02  8.294e+01   2.205 0.027597 *  
## YrSold         -1.486e+03  8.603e+02  -1.727 0.084323 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 42590 on 1430 degrees of freedom
## Multiple R-squared:  0.7183, Adjusted R-squared:  0.7126 
## F-statistic: 125.8 on 29 and 1430 DF,  p-value: < 2.2e-16
imcdiag(lm_5.lm)
## 
## Call:
## imcdiag(mod = lm_5.lm)
## 
## 
## All Individual Multicollinearity Diagnostics Result
## 
##                   VIF    TOL      Wi      Fi Leamer    CVIF Klein   IND1   IND2
## BedroomAbvGr   1.4940 0.6693 25.2490 26.2025 0.8181 -0.2998     0 0.0131 1.6610
## BsmtFinType2_q 1.1075 0.9029  5.4932  5.7006 0.9502 -0.2223     0 0.0177 0.4875
## BsmtFullBath   1.2515 0.7990 12.8531 13.3385 0.8939 -0.2512     0 0.0156 1.0094
## BsmtHalfBath   1.0826 0.9237  4.2205  4.3799 0.9611 -0.2173     0 0.0181 0.3832
## Electrical_q   1.1823 0.8458  9.3193  9.6712 0.9197 -0.2373     0 0.0165 0.7747
## EnclosedPorch  1.1191 0.8936  6.0846  6.3144 0.9453 -0.2246     0 0.0175 0.5344
## Fence_q        1.1582 0.8634  8.0854  8.3907 0.9292 -0.2324     0 0.0169 0.6861
## Fireplaces     1.3347 0.7492 17.1067 17.7527 0.8656 -0.2679     0 0.0147 1.2597
## Foundation_q   1.5690 0.6374 29.0775 30.1755 0.7984 -0.3149     0 0.0125 1.8215
## FullBath       2.1337 0.4687 57.9386 60.1265 0.6846 -0.4282     0 0.0092 2.6688
## HalfBath       1.2956 0.7719 15.1057 15.6761 0.8786 -0.2600     0 0.0151 1.1459
## HeatingQC_q    1.6222 0.6164 31.8004 33.0013 0.7851 -0.3256     0 0.0121 1.9266
## KitchenAbvGr   1.3290 0.7524 16.8147 17.4497 0.8674 -0.2667     0 0.0147 1.2435
## KitchenQual_q  2.0714 0.4828 54.7556 56.8233 0.6948 -0.4157     0 0.0094 2.5980
## LotArea        1.1842 0.8444  9.4145  9.7700 0.9189 -0.2377     0 0.0165 0.7813
## LotFrontage    1.1481 0.8710  7.5678  7.8535 0.9333 -0.2304     0 0.0170 0.6479
## LowQualFinSF   1.0441 0.9578  2.2526  2.3377 0.9787 -0.2095     0 0.0187 0.2120
## MasVnrArea     1.2160 0.8224 11.0368 11.4536 0.9069 -0.2440     0 0.0161 0.8921
## MiscVal        1.0260 0.9747  1.3288  1.3790 0.9872 -0.2059     0 0.0191 0.1273
## MoSold         1.0437 0.9582  2.2317  2.3160 0.9789 -0.2095     0 0.0187 0.2102
## MSSubClass     1.3024 0.7678 15.4542 16.0378 0.8763 -0.2614     0 0.0150 1.1662
## OpenPorchSF    1.1661 0.8575  8.4901  8.8107 0.9260 -0.2340     0 0.0168 0.7156
## OverallCond    1.2988 0.7700 15.2701 15.8467 0.8775 -0.2607     0 0.0151 1.1555
## PoolArea       1.0874 0.9196  4.4681  4.6368 0.9590 -0.2182     0 0.0180 0.4038
## ScreenPorch    1.0920 0.9157  4.7039  4.8816 0.9569 -0.2192     0 0.0179 0.4234
## WoodDeckSF     1.1978 0.8348 10.1113 10.4932 0.9137 -0.2404     0 0.0163 0.8296
## X3SsnPorch     1.0204 0.9800  1.0429  1.0823 0.9900 -0.2048     0 0.0192 0.1004
## YearRemodAdd   2.3590 0.4239 69.4554 72.0782 0.6511 -0.4734     0 0.0083 2.8937
## YrSold         1.0503 0.9521  2.5723  2.6695 0.9757 -0.2108     0 0.0186 0.2407
## 
## 1 --> COLLINEARITY is detected by the test 
## 0 --> COLLINEARITY is not detected by the test
## 
## BedroomAbvGr , BsmtFinType2_q , BsmtHalfBath , Electrical_q , EnclosedPorch , KitchenAbvGr , LowQualFinSF , MiscVal , MoSold , PoolArea , X3SsnPorch , YrSold , coefficient(s) are non-significant may be due to multicollinearity
## 
## R-square of y on all x: 0.7183 
## 
## * use method argument to check which regressors may be the reason of collinearity
## ===================================

Version 5 Discussion :

Note that after computing Linear Model V5, we have Multiple - \(R^{2} = 0.7183\) and \(R^{2} = 0.7126\) - A Decline - Not in the Expected Direction


Final Model Adjustments

We will test our model with a low score stepAIC model

stepAIC is one of the most commonly used search method for feature selection. We try to keep on minimizing the stepAIC value to come up with the final set of features. “stepAIC” does not necessarily mean to improve the model performance, however, it is used to simplify the model without impacting much on the performance.

We will use the stepAIC procedure to determine the final model components !

This is based on a model prediction with the lowest AIC score :

the scores for our model range from AIC=31154.79 to AIC=31141.34

We will be using the model with score : AIC=31141.34

#### - StepAIc

stepAIC(lm_5.lm, direction="both")
## Start:  AIC=31154.79
## SalePrice ~ BedroomAbvGr + BsmtFinType2_q + BsmtFullBath + BsmtHalfBath + 
##     Electrical_q + EnclosedPorch + Fence_q + Fireplaces + Foundation_q + 
##     FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + 
##     LotArea + LotFrontage + LowQualFinSF + MasVnrArea + MiscVal + 
##     MoSold + MSSubClass + OpenPorchSF + OverallCond + PoolArea + 
##     ScreenPorch + WoodDeckSF + X3SsnPorch + YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - BsmtFinType2_q  1 5.6243e+05 2.5934e+12 31153
## - MoSold          1 4.6398e+07 2.5935e+12 31153
## - Electrical_q    1 1.8673e+08 2.5936e+12 31153
## - MiscVal         1 6.1561e+08 2.5940e+12 31153
## - BsmtHalfBath    1 6.6417e+08 2.5941e+12 31153
## - PoolArea        1 9.3916e+08 2.5944e+12 31153
## - X3SsnPorch      1 1.5586e+09 2.5950e+12 31154
## - EnclosedPorch   1 1.6594e+09 2.5951e+12 31154
## - BedroomAbvGr    1 1.9389e+09 2.5954e+12 31154
## <none>                         2.5934e+12 31155
## - LowQualFinSF    1 4.3289e+09 2.5977e+12 31155
## - KitchenAbvGr    1 4.7129e+09 2.5981e+12 31155
## - YrSold          1 5.4112e+09 2.5988e+12 31156
## - OverallCond     1 8.4800e+09 2.6019e+12 31158
## - Fence_q         1 8.8155e+09 2.6022e+12 31158
## - YearRemodAdd    1 8.8194e+09 2.6022e+12 31158
## - HeatingQC_q     1 1.0119e+10 2.6035e+12 31159
## - OpenPorchSF     1 1.4778e+10 2.6082e+12 31161
## - ScreenPorch     1 2.2807e+10 2.6162e+12 31166
## - Foundation_q    1 3.1514e+10 2.6249e+12 31170
## - LotFrontage     1 3.9923e+10 2.6333e+12 31175
## - WoodDeckSF      1 5.1709e+10 2.6451e+12 31182
## - LotArea         1 5.2519e+10 2.6459e+12 31182
## - MSSubClass      1 6.6715e+10 2.6601e+12 31190
## - HalfBath        1 7.0554e+10 2.6640e+12 31192
## - BsmtFullBath    1 1.1284e+11 2.7062e+12 31215
## - Fireplaces      1 1.8599e+11 2.7794e+12 31254
## - FullBath        1 2.4791e+11 2.8413e+12 31286
## - MasVnrArea      1 2.8094e+11 2.8743e+12 31303
## - KitchenQual_q   1 3.7518e+11 2.9686e+12 31350
## 
## Step:  AIC=31152.79
## SalePrice ~ BedroomAbvGr + BsmtFullBath + BsmtHalfBath + Electrical_q + 
##     EnclosedPorch + Fence_q + Fireplaces + Foundation_q + FullBath + 
##     HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + 
##     LotFrontage + LowQualFinSF + MasVnrArea + MiscVal + MoSold + 
##     MSSubClass + OpenPorchSF + OverallCond + PoolArea + ScreenPorch + 
##     WoodDeckSF + X3SsnPorch + YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - MoSold          1 4.6241e+07 2.5935e+12 31151
## - Electrical_q    1 1.8620e+08 2.5936e+12 31151
## - MiscVal         1 6.1617e+08 2.5940e+12 31151
## - BsmtHalfBath    1 6.7729e+08 2.5941e+12 31151
## - PoolArea        1 9.3863e+08 2.5944e+12 31151
## - X3SsnPorch      1 1.5584e+09 2.5950e+12 31152
## - EnclosedPorch   1 1.6676e+09 2.5951e+12 31152
## - BedroomAbvGr    1 1.9388e+09 2.5954e+12 31152
## <none>                         2.5934e+12 31153
## - LowQualFinSF    1 4.3299e+09 2.5977e+12 31153
## - KitchenAbvGr    1 4.7388e+09 2.5982e+12 31154
## - YrSold          1 5.4112e+09 2.5988e+12 31154
## + BsmtFinType2_q  1 5.6243e+05 2.5934e+12 31155
## - OverallCond     1 8.4924e+09 2.6019e+12 31156
## - YearRemodAdd    1 8.8195e+09 2.6022e+12 31156
## - Fence_q         1 8.8855e+09 2.6023e+12 31156
## - HeatingQC_q     1 1.0126e+10 2.6035e+12 31157
## - OpenPorchSF     1 1.4781e+10 2.6082e+12 31159
## - ScreenPorch     1 2.2847e+10 2.6163e+12 31164
## - Foundation_q    1 3.1694e+10 2.6251e+12 31169
## - LotFrontage     1 3.9923e+10 2.6333e+12 31173
## - WoodDeckSF      1 5.1831e+10 2.6452e+12 31180
## - LotArea         1 5.2676e+10 2.6461e+12 31180
## - MSSubClass      1 6.6735e+10 2.6601e+12 31188
## - HalfBath        1 7.0555e+10 2.6640e+12 31190
## - BsmtFullBath    1 1.1601e+11 2.7094e+12 31215
## - Fireplaces      1 1.8601e+11 2.7794e+12 31252
## - FullBath        1 2.4791e+11 2.8413e+12 31284
## - MasVnrArea      1 2.8143e+11 2.8748e+12 31301
## - KitchenQual_q   1 3.7526e+11 2.9687e+12 31348
## 
## Step:  AIC=31150.82
## SalePrice ~ BedroomAbvGr + BsmtFullBath + BsmtHalfBath + Electrical_q + 
##     EnclosedPorch + Fence_q + Fireplaces + Foundation_q + FullBath + 
##     HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + 
##     LotFrontage + LowQualFinSF + MasVnrArea + MiscVal + MSSubClass + 
##     OpenPorchSF + OverallCond + PoolArea + ScreenPorch + WoodDeckSF + 
##     X3SsnPorch + YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - Electrical_q    1 1.8698e+08 2.5936e+12 31149
## - MiscVal         1 6.1528e+08 2.5941e+12 31149
## - BsmtHalfBath    1 6.8627e+08 2.5941e+12 31149
## - PoolArea        1 9.1889e+08 2.5944e+12 31149
## - X3SsnPorch      1 1.5787e+09 2.5950e+12 31150
## - EnclosedPorch   1 1.6544e+09 2.5951e+12 31150
## - BedroomAbvGr    1 1.9580e+09 2.5954e+12 31150
## <none>                         2.5935e+12 31151
## - LowQualFinSF    1 4.3066e+09 2.5978e+12 31151
## - KitchenAbvGr    1 4.7098e+09 2.5982e+12 31152
## - YrSold          1 5.6781e+09 2.5991e+12 31152
## + MoSold          1 4.6241e+07 2.5934e+12 31153
## + BsmtFinType2_q  1 4.0577e+05 2.5935e+12 31153
## - OverallCond     1 8.4819e+09 2.6019e+12 31154
## - YearRemodAdd    1 8.8201e+09 2.6023e+12 31154
## - Fence_q         1 8.8690e+09 2.6023e+12 31154
## - HeatingQC_q     1 1.0112e+10 2.6036e+12 31155
## - OpenPorchSF     1 1.4917e+10 2.6084e+12 31157
## - ScreenPorch     1 2.2894e+10 2.6164e+12 31162
## - Foundation_q    1 3.1651e+10 2.6251e+12 31167
## - LotFrontage     1 3.9948e+10 2.6334e+12 31171
## - WoodDeckSF      1 5.1925e+10 2.6454e+12 31178
## - LotArea         1 5.2632e+10 2.6461e+12 31178
## - MSSubClass      1 6.6792e+10 2.6602e+12 31186
## - HalfBath        1 7.0513e+10 2.6640e+12 31188
## - BsmtFullBath    1 1.1596e+11 2.7094e+12 31213
## - Fireplaces      1 1.8644e+11 2.7799e+12 31250
## - FullBath        1 2.4800e+11 2.8415e+12 31282
## - MasVnrArea      1 2.8155e+11 2.8750e+12 31299
## - KitchenQual_q   1 3.7648e+11 2.9699e+12 31347
## 
## Step:  AIC=31148.92
## SalePrice ~ BedroomAbvGr + BsmtFullBath + BsmtHalfBath + EnclosedPorch + 
##     Fence_q + Fireplaces + Foundation_q + FullBath + HalfBath + 
##     HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + 
##     LowQualFinSF + MasVnrArea + MiscVal + MSSubClass + OpenPorchSF + 
##     OverallCond + PoolArea + ScreenPorch + WoodDeckSF + X3SsnPorch + 
##     YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - MiscVal         1 5.9413e+08 2.5942e+12 31147
## - BsmtHalfBath    1 6.6863e+08 2.5943e+12 31147
## - PoolArea        1 9.2519e+08 2.5946e+12 31147
## - X3SsnPorch      1 1.5840e+09 2.5952e+12 31148
## - EnclosedPorch   1 1.7541e+09 2.5954e+12 31148
## - BedroomAbvGr    1 1.8876e+09 2.5955e+12 31148
## <none>                         2.5936e+12 31149
## - LowQualFinSF    1 4.3109e+09 2.5980e+12 31149
## - KitchenAbvGr    1 4.5488e+09 2.5982e+12 31150
## - YrSold          1 5.7366e+09 2.5994e+12 31150
## + Electrical_q    1 1.8698e+08 2.5935e+12 31151
## + MoSold          1 4.7017e+07 2.5936e+12 31151
## + BsmtFinType2_q  1 3.8670e+03 2.5936e+12 31151
## - OverallCond     1 8.3354e+09 2.6020e+12 31152
## - YearRemodAdd    1 8.6341e+09 2.6023e+12 31152
## - Fence_q         1 9.0162e+09 2.6027e+12 31152
## - HeatingQC_q     1 1.0104e+10 2.6037e+12 31153
## - OpenPorchSF     1 1.4897e+10 2.6085e+12 31155
## - ScreenPorch     1 2.2919e+10 2.6166e+12 31160
## - Foundation_q    1 3.1586e+10 2.6252e+12 31165
## - LotFrontage     1 4.0029e+10 2.6337e+12 31169
## - WoodDeckSF      1 5.1802e+10 2.6454e+12 31176
## - LotArea         1 5.2627e+10 2.6463e+12 31176
## - MSSubClass      1 6.7306e+10 2.6610e+12 31184
## - HalfBath        1 7.0434e+10 2.6641e+12 31186
## - BsmtFullBath    1 1.1610e+11 2.7097e+12 31211
## - Fireplaces      1 1.8629e+11 2.7799e+12 31248
## - FullBath        1 2.4782e+11 2.8415e+12 31280
## - MasVnrArea      1 2.8145e+11 2.8751e+12 31297
## - KitchenQual_q   1 3.7635e+11 2.9700e+12 31345
## 
## Step:  AIC=31147.26
## SalePrice ~ BedroomAbvGr + BsmtFullBath + BsmtHalfBath + EnclosedPorch + 
##     Fence_q + Fireplaces + Foundation_q + FullBath + HalfBath + 
##     HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + 
##     LowQualFinSF + MasVnrArea + MSSubClass + OpenPorchSF + OverallCond + 
##     PoolArea + ScreenPorch + WoodDeckSF + X3SsnPorch + YearRemodAdd + 
##     YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - BsmtHalfBath    1 6.4343e+08 2.5949e+12 31146
## - PoolArea        1 9.8078e+08 2.5952e+12 31146
## - X3SsnPorch      1 1.5822e+09 2.5958e+12 31146
## - EnclosedPorch   1 1.7949e+09 2.5960e+12 31146
## - BedroomAbvGr    1 1.8535e+09 2.5961e+12 31146
## <none>                         2.5942e+12 31147
## - LowQualFinSF    1 4.3018e+09 2.5985e+12 31148
## - KitchenAbvGr    1 4.3334e+09 2.5986e+12 31148
## - YrSold          1 5.7461e+09 2.6000e+12 31149
## + MiscVal         1 5.9413e+08 2.5936e+12 31149
## + Electrical_q    1 1.6582e+08 2.5941e+12 31149
## + MoSold          1 4.6090e+07 2.5942e+12 31149
## + BsmtFinType2_q  1 1.6124e+05 2.5942e+12 31149
## - OverallCond     1 8.6709e+09 2.6029e+12 31150
## - YearRemodAdd    1 8.7414e+09 2.6030e+12 31150
## - Fence_q         1 8.9307e+09 2.6032e+12 31150
## - HeatingQC_q     1 1.0118e+10 2.6044e+12 31151
## - OpenPorchSF     1 1.4833e+10 2.6091e+12 31154
## - ScreenPorch     1 2.3212e+10 2.6175e+12 31158
## - Foundation_q    1 3.1933e+10 2.6262e+12 31163
## - LotFrontage     1 3.9556e+10 2.6338e+12 31167
## - WoodDeckSF      1 5.1747e+10 2.6460e+12 31174
## - LotArea         1 5.3262e+10 2.6475e+12 31175
## - MSSubClass      1 6.7865e+10 2.6621e+12 31183
## - HalfBath        1 7.0716e+10 2.6650e+12 31185
## - BsmtFullBath    1 1.1578e+11 2.7100e+12 31209
## - Fireplaces      1 1.8644e+11 2.7807e+12 31247
## - FullBath        1 2.4789e+11 2.8421e+12 31279
## - MasVnrArea      1 2.8129e+11 2.8755e+12 31296
## - KitchenQual_q   1 3.7579e+11 2.9700e+12 31343
## 
## Step:  AIC=31145.62
## SalePrice ~ BedroomAbvGr + BsmtFullBath + EnclosedPorch + Fence_q + 
##     Fireplaces + Foundation_q + FullBath + HalfBath + HeatingQC_q + 
##     KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + 
##     MasVnrArea + MSSubClass + OpenPorchSF + OverallCond + PoolArea + 
##     ScreenPorch + WoodDeckSF + X3SsnPorch + YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - PoolArea        1 1.0159e+09 2.5959e+12 31144
## - X3SsnPorch      1 1.6705e+09 2.5966e+12 31145
## - EnclosedPorch   1 1.7544e+09 2.5966e+12 31145
## - BedroomAbvGr    1 1.9922e+09 2.5969e+12 31145
## <none>                         2.5949e+12 31146
## - LowQualFinSF    1 4.2366e+09 2.5991e+12 31146
## - KitchenAbvGr    1 4.4287e+09 2.5993e+12 31146
## - YrSold          1 5.9091e+09 2.6008e+12 31147
## + BsmtHalfBath    1 6.4343e+08 2.5942e+12 31147
## + MiscVal         1 5.6894e+08 2.5943e+12 31147
## + Electrical_q    1 1.4997e+08 2.5947e+12 31148
## + MoSold          1 5.4750e+07 2.5948e+12 31148
## + BsmtFinType2_q  1 1.0796e+07 2.5949e+12 31148
## - YearRemodAdd    1 8.9424e+09 2.6038e+12 31149
## - Fence_q         1 8.9643e+09 2.6038e+12 31149
## - OverallCond     1 9.0629e+09 2.6039e+12 31149
## - HeatingQC_q     1 9.9983e+09 2.6049e+12 31149
## - OpenPorchSF     1 1.4786e+10 2.6097e+12 31152
## - ScreenPorch     1 2.3409e+10 2.6183e+12 31157
## - Foundation_q    1 3.1816e+10 2.6267e+12 31161
## - LotFrontage     1 3.9353e+10 2.6342e+12 31166
## - WoodDeckSF      1 5.2621e+10 2.6475e+12 31173
## - LotArea         1 5.4133e+10 2.6490e+12 31174
## - MSSubClass      1 6.7445e+10 2.6623e+12 31181
## - HalfBath        1 7.0207e+10 2.6651e+12 31183
## - BsmtFullBath    1 1.1618e+11 2.7111e+12 31208
## - Fireplaces      1 1.8736e+11 2.7822e+12 31245
## - FullBath        1 2.4761e+11 2.8425e+12 31277
## - MasVnrArea      1 2.8349e+11 2.8784e+12 31295
## - KitchenQual_q   1 3.7585e+11 2.9707e+12 31341
## 
## Step:  AIC=31144.19
## SalePrice ~ BedroomAbvGr + BsmtFullBath + EnclosedPorch + Fence_q + 
##     Fireplaces + Foundation_q + FullBath + HalfBath + HeatingQC_q + 
##     KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + 
##     MasVnrArea + MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + 
##     WoodDeckSF + X3SsnPorch + YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - X3SsnPorch      1 1.6690e+09 2.5976e+12 31143
## - EnclosedPorch   1 1.9091e+09 2.5978e+12 31143
## - BedroomAbvGr    1 2.0693e+09 2.5980e+12 31143
## <none>                         2.5959e+12 31144
## - LowQualFinSF    1 4.4151e+09 2.6003e+12 31145
## - KitchenAbvGr    1 4.5340e+09 2.6004e+12 31145
## + PoolArea        1 1.0159e+09 2.5949e+12 31146
## - YrSold          1 6.2523e+09 2.6022e+12 31146
## + BsmtHalfBath    1 6.7857e+08 2.5952e+12 31146
## + MiscVal         1 6.2370e+08 2.5953e+12 31146
## + Electrical_q    1 1.5454e+08 2.5957e+12 31146
## + MoSold          1 3.2209e+07 2.5959e+12 31146
## + BsmtFinType2_q  1 6.0394e+06 2.5959e+12 31146
## - Fence_q         1 8.2796e+09 2.6042e+12 31147
## - YearRemodAdd    1 8.9360e+09 2.6048e+12 31147
## - OverallCond     1 8.9848e+09 2.6049e+12 31147
## - HeatingQC_q     1 9.5788e+09 2.6055e+12 31148
## - OpenPorchSF     1 1.5029e+10 2.6109e+12 31151
## - ScreenPorch     1 2.3823e+10 2.6197e+12 31156
## - Foundation_q    1 3.1964e+10 2.6279e+12 31160
## - LotFrontage     1 4.1127e+10 2.6370e+12 31165
## - WoodDeckSF      1 5.3365e+10 2.6493e+12 31172
## - LotArea         1 5.4778e+10 2.6507e+12 31173
## - MSSubClass      1 6.6787e+10 2.6627e+12 31179
## - HalfBath        1 7.0110e+10 2.6660e+12 31181
## - BsmtFullBath    1 1.1744e+11 2.7133e+12 31207
## - Fireplaces      1 1.8887e+11 2.7848e+12 31245
## - FullBath        1 2.4831e+11 2.8442e+12 31276
## - MasVnrArea      1 2.8274e+11 2.8786e+12 31293
## - KitchenQual_q   1 3.7885e+11 2.9747e+12 31341
## 
## Step:  AIC=31143.13
## SalePrice ~ BedroomAbvGr + BsmtFullBath + EnclosedPorch + Fence_q + 
##     Fireplaces + Foundation_q + FullBath + HalfBath + HeatingQC_q + 
##     KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + 
##     MasVnrArea + MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + 
##     WoodDeckSF + YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - EnclosedPorch   1 1.7937e+09 2.5994e+12 31142
## - BedroomAbvGr    1 1.9290e+09 2.5995e+12 31142
## <none>                         2.5976e+12 31143
## - LowQualFinSF    1 4.4558e+09 2.6020e+12 31144
## - KitchenAbvGr    1 4.5787e+09 2.6021e+12 31144
## + X3SsnPorch      1 1.6690e+09 2.5959e+12 31144
## + PoolArea        1 1.0144e+09 2.5966e+12 31145
## - YrSold          1 6.1546e+09 2.6037e+12 31145
## + BsmtHalfBath    1 7.6899e+08 2.5968e+12 31145
## + MiscVal         1 6.2007e+08 2.5969e+12 31145
## + Electrical_q    1 1.5842e+08 2.5974e+12 31145
## + MoSold          1 5.0677e+07 2.5975e+12 31145
## + BsmtFinType2_q  1 5.8051e+06 2.5976e+12 31145
## - Fence_q         1 8.1271e+09 2.6057e+12 31146
## - YearRemodAdd    1 8.9260e+09 2.6065e+12 31146
## - OverallCond     1 9.3505e+09 2.6069e+12 31146
## - HeatingQC_q     1 9.8788e+09 2.6074e+12 31147
## - OpenPorchSF     1 1.4801e+10 2.6124e+12 31149
## - ScreenPorch     1 2.3379e+10 2.6209e+12 31154
## - Foundation_q    1 3.2527e+10 2.6301e+12 31159
## - LotFrontage     1 4.1319e+10 2.6389e+12 31164
## - WoodDeckSF      1 5.2433e+10 2.6500e+12 31170
## - LotArea         1 5.5232e+10 2.6528e+12 31172
## - MSSubClass      1 6.7759e+10 2.6653e+12 31179
## - HalfBath        1 7.0324e+10 2.6679e+12 31180
## - BsmtFullBath    1 1.1751e+11 2.7151e+12 31206
## - Fireplaces      1 1.8924e+11 2.7868e+12 31244
## - FullBath        1 2.5042e+11 2.8480e+12 31276
## - MasVnrArea      1 2.8345e+11 2.8810e+12 31292
## - KitchenQual_q   1 3.7784e+11 2.9754e+12 31339
## 
## Step:  AIC=31142.14
## SalePrice ~ BedroomAbvGr + BsmtFullBath + Fence_q + Fireplaces + 
##     Foundation_q + FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + 
##     KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + MasVnrArea + 
##     MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + WoodDeckSF + 
##     YearRemodAdd + YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## - BedroomAbvGr    1 2.1315e+09 2.6015e+12 31141
## <none>                         2.5994e+12 31142
## - KitchenAbvGr    1 4.5631e+09 2.6039e+12 31143
## - LowQualFinSF    1 4.6323e+09 2.6040e+12 31143
## + EnclosedPorch   1 1.7937e+09 2.5976e+12 31143
## + X3SsnPorch      1 1.5535e+09 2.5978e+12 31143
## + PoolArea        1 1.1642e+09 2.5982e+12 31144
## - YrSold          1 6.1589e+09 2.6055e+12 31144
## + BsmtHalfBath    1 7.2368e+08 2.5986e+12 31144
## + MiscVal         1 6.6696e+08 2.5987e+12 31144
## + Electrical_q    1 2.5250e+08 2.5991e+12 31144
## + MoSold          1 3.4117e+07 2.5993e+12 31144
## + BsmtFinType2_q  1 1.9561e+07 2.5993e+12 31144
## - Fence_q         1 7.9341e+09 2.6073e+12 31145
## - YearRemodAdd    1 8.1863e+09 2.6075e+12 31145
## - OverallCond     1 9.6087e+09 2.6090e+12 31146
## - HeatingQC_q     1 1.0043e+10 2.6094e+12 31146
## - OpenPorchSF     1 1.4414e+10 2.6138e+12 31148
## - ScreenPorch     1 2.2229e+10 2.6216e+12 31153
## - Foundation_q    1 3.1218e+10 2.6306e+12 31158
## - LotFrontage     1 4.2054e+10 2.6414e+12 31164
## - WoodDeckSF      1 5.1107e+10 2.6505e+12 31169
## - LotArea         1 5.5078e+10 2.6544e+12 31171
## - MSSubClass      1 6.7268e+10 2.6666e+12 31177
## - HalfBath        1 6.9432e+10 2.6688e+12 31179
## - BsmtFullBath    1 1.1720e+11 2.7166e+12 31205
## - Fireplaces      1 1.9137e+11 2.7907e+12 31244
## - FullBath        1 2.4973e+11 2.8491e+12 31274
## - MasVnrArea      1 2.8181e+11 2.8812e+12 31290
## - KitchenQual_q   1 3.8328e+11 2.9826e+12 31341
## 
## Step:  AIC=31141.34
## SalePrice ~ BsmtFullBath + Fence_q + Fireplaces + Foundation_q + 
##     FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + 
##     LotArea + LotFrontage + LowQualFinSF + MasVnrArea + MSSubClass + 
##     OpenPorchSF + OverallCond + ScreenPorch + WoodDeckSF + YearRemodAdd + 
##     YrSold
## 
##                  Df  Sum of Sq        RSS   AIC
## <none>                         2.6015e+12 31141
## - KitchenAbvGr    1 3.6961e+09 2.6052e+12 31141
## + BedroomAbvGr    1 2.1315e+09 2.5994e+12 31142
## + EnclosedPorch   1 1.9962e+09 2.5995e+12 31142
## - LowQualFinSF    1 5.3295e+09 2.6068e+12 31142
## + X3SsnPorch      1 1.4055e+09 2.6001e+12 31143
## + PoolArea        1 1.2570e+09 2.6002e+12 31143
## + BsmtHalfBath    1 8.7035e+08 2.6006e+12 31143
## - YrSold          1 6.3919e+09 2.6079e+12 31143
## + MiscVal         1 6.3019e+08 2.6009e+12 31143
## + Electrical_q    1 1.6847e+08 2.6013e+12 31143
## + MoSold          1 4.9617e+07 2.6014e+12 31143
## + BsmtFinType2_q  1 2.3554e+07 2.6015e+12 31143
## - YearRemodAdd    1 7.2376e+09 2.6087e+12 31143
## - Fence_q         1 7.2819e+09 2.6088e+12 31143
## - HeatingQC_q     1 9.8954e+09 2.6114e+12 31145
## - OverallCond     1 1.0868e+10 2.6124e+12 31145
## - OpenPorchSF     1 1.4411e+10 2.6159e+12 31147
## - ScreenPorch     1 2.2514e+10 2.6240e+12 31152
## - Foundation_q    1 3.0523e+10 2.6320e+12 31156
## - LotFrontage     1 4.4215e+10 2.6457e+12 31164
## - WoodDeckSF      1 5.1393e+10 2.6529e+12 31168
## - LotArea         1 5.6539e+10 2.6580e+12 31171
## - MSSubClass      1 7.2283e+10 2.6738e+12 31179
## - HalfBath        1 8.2888e+10 2.6844e+12 31185
## - BsmtFullBath    1 1.1551e+11 2.7170e+12 31203
## - Fireplaces      1 1.9096e+11 2.7925e+12 31243
## - MasVnrArea      1 2.8291e+11 2.8844e+12 31290
## - FullBath        1 3.2009e+11 2.9216e+12 31309
## - KitchenQual_q   1 3.8128e+11 2.9828e+12 31339
## 
## Call:
## lm(formula = SalePrice ~ BsmtFullBath + Fence_q + Fireplaces + 
##     Foundation_q + FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + 
##     KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + MasVnrArea + 
##     MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + WoodDeckSF + 
##     YearRemodAdd + YrSold, data = p2_train)
## 
## Coefficients:
##   (Intercept)   BsmtFullBath        Fence_q     Fireplaces   Foundation_q  
##     2.845e+06      1.850e+04     -1.954e+03      2.041e+04     -7.830e+03  
##      FullBath       HalfBath    HeatingQC_q   KitchenAbvGr  KitchenQual_q  
##     3.577e+04      1.632e+04      3.442e+03     -8.099e+03      3.467e+04  
##       LotArea    LotFrontage   LowQualFinSF     MasVnrArea     MSSubClass  
##     6.739e-01      1.679e+02      3.988e+01      8.451e+01     -1.869e+02  
##   OpenPorchSF    OverallCond    ScreenPorch     WoodDeckSF   YearRemodAdd  
##     5.106e+01      2.748e+03      7.291e+01      5.129e+01      1.611e+02  
##        YrSold  
##    -1.592e+03

Final Model

This model is the result of 5 iterations of the original model followed by a stepAIC computation producing the Final Model as follows :

## The following model computed the lowest score of AIC=31141.34

lm_final.lm <- lm((SalePrice ~ BsmtFullBath + Fence_q + Fireplaces + Foundation_q + FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + MasVnrArea + MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + WoodDeckSF + YearRemodAdd + YrSold), data = p2_train)

summary(lm_final.lm)
## 
## Call:
## lm(formula = (SalePrice ~ BsmtFullBath + Fence_q + Fireplaces + 
##     Foundation_q + FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + 
##     KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + MasVnrArea + 
##     MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + WoodDeckSF + 
##     YearRemodAdd + YrSold), data = p2_train)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -341177  -24710   -3586   19167  391779 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    2.845e+06  1.700e+06   1.674  0.09434 .  
## BsmtFullBath   1.850e+04  2.314e+03   7.993 2.67e-15 ***
## Fence_q       -1.954e+03  9.734e+02  -2.007  0.04494 *  
## Fireplaces     2.041e+04  1.986e+03  10.278  < 2e-16 ***
## Foundation_q  -7.830e+03  1.906e+03  -4.109 4.20e-05 ***
## FullBath       3.577e+04  2.688e+03  13.306  < 2e-16 ***
## HalfBath       1.632e+04  2.411e+03   6.771 1.86e-11 ***
## HeatingQC_q    3.442e+03  1.471e+03   2.340  0.01944 *  
## KitchenAbvGr  -8.099e+03  5.664e+03  -1.430  0.15298    
## KitchenQual_q  3.467e+04  2.388e+03  14.523  < 2e-16 ***
## LotArea        6.739e-01  1.205e-01   5.592 2.68e-08 ***
## LotFrontage    1.679e+02  3.396e+01   4.945 8.49e-07 ***
## LowQualFinSF   3.988e+01  2.323e+01   1.717  0.08620 .  
## MasVnrArea     8.451e+01  6.755e+00  12.510  < 2e-16 ***
## MSSubClass    -1.869e+02  2.956e+01  -6.323 3.41e-10 ***
## OpenPorchSF    5.106e+01  1.808e+01   2.823  0.00482 ** 
## OverallCond    2.748e+03  1.121e+03   2.452  0.01433 *  
## ScreenPorch    7.291e+01  2.066e+01   3.529  0.00043 ***
## WoodDeckSF     5.129e+01  9.620e+00   5.332 1.13e-07 ***
## YearRemodAdd   1.611e+02  8.051e+01   2.001  0.04560 *  
## YrSold        -1.592e+03  8.464e+02  -1.880  0.06027 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 42520 on 1439 degrees of freedom
## Multiple R-squared:  0.7175, Adjusted R-squared:  0.7135 
## F-statistic: 182.7 on 20 and 1439 DF,  p-value: < 2.2e-16

Residuals Discussion

The histogram of the residuals shows an almost perfect normal distribution with mean approximately = 0

In the QQ lot, data that aligns closely to the red line indicates a normal distribution. If the points skew drastically from the line, you could consider adjusting your model by adding or removing other variables in the regression model, this model is the result of that model adjustment.

hist(lm_final.lm$residuals, prob = TRUE)
abline(v = mean(lm_final.lm$residuals),                       # Add line for mean
       col = "red",
       lwd = 3)
lines(density(lm_final.lm$residuals),col = "blue")

qqnorm(lm_final.lm$residuals)
qqline(lm_final.lm$residuals, col = "red")

The fitted and residual values seem to have a linear relationship, there is some evidence of heteroskedastic behavior

plot(lm_final.lm$fitted.values, lm_final.lm$residuals, 
     xlab="Fitted Values", ylab="Residuals",
     main="Residuals Plot",col = "blue")
abline(h=0)


Predicting the Test Data

p2_test %>% select(order(colnames(p2_test)))
str(p2_test)
## 'data.frame':    1459 obs. of  88 variables:
##  $ Id            : int  1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 ...
##  $ MSSubClass    : int  20 20 60 60 120 60 20 60 20 20 ...
##  $ MSZoning      : chr  "RH" "RL" "RL" "RL" ...
##  $ LotFrontage   : num  80 81 74 78 43 75 0 63 85 70 ...
##  $ LotArea       : int  11622 14267 13830 9978 5005 10000 7980 8402 10176 8400 ...
##  $ Street        : chr  "Pave" "Pave" "Pave" "Pave" ...
##  $ Alley         : chr  "0" "0" "0" "0" ...
##  $ LotShape      : chr  "Reg" "IR1" "IR1" "IR1" ...
##  $ LandContour   : chr  "Lvl" "Lvl" "Lvl" "Lvl" ...
##  $ Utilities     : chr  "AllPub" "AllPub" "AllPub" "AllPub" ...
##  $ LotConfig     : chr  "Inside" "Corner" "Inside" "Inside" ...
##  $ LandSlope     : chr  "Gtl" "Gtl" "Gtl" "Gtl" ...
##  $ Neighborhood  : chr  "NAmes" "NAmes" "Gilbert" "Gilbert" ...
##  $ Condition1    : chr  "Feedr" "Norm" "Norm" "Norm" ...
##  $ Condition2    : chr  "Norm" "Norm" "Norm" "Norm" ...
##  $ BldgType      : chr  "1Fam" "1Fam" "1Fam" "1Fam" ...
##  $ HouseStyle    : chr  "1Story" "1Story" "2Story" "2Story" ...
##  $ OverallQual   : int  5 6 5 6 8 6 6 6 7 4 ...
##  $ OverallCond   : int  6 6 5 6 5 5 7 5 5 5 ...
##  $ YearBuilt     : int  1961 1958 1997 1998 1992 1993 1992 1998 1990 1970 ...
##  $ YearRemodAdd  : int  1961 1958 1998 1998 1992 1994 2007 1998 1990 1970 ...
##  $ RoofStyle     : chr  "Gable" "Hip" "Gable" "Gable" ...
##  $ RoofMatl      : chr  "CompShg" "CompShg" "CompShg" "CompShg" ...
##  $ Exterior1st   : chr  "VinylSd" "Wd Sdng" "VinylSd" "VinylSd" ...
##  $ Exterior2nd   : chr  "VinylSd" "Wd Sdng" "VinylSd" "VinylSd" ...
##  $ MasVnrType    : chr  "None" "BrkFace" "None" "BrkFace" ...
##  $ MasVnrArea    : num  0 108 0 20 0 0 0 0 0 0 ...
##  $ ExterQual     : chr  "TA" "TA" "TA" "TA" ...
##  $ ExterCond     : chr  "TA" "TA" "TA" "TA" ...
##  $ Foundation    : chr  "CBlock" "CBlock" "PConc" "PConc" ...
##  $ BsmtQual      : chr  "TA" "TA" "Gd" "TA" ...
##  $ BsmtCond      : chr  "TA" "TA" "TA" "TA" ...
##  $ BsmtExposure  : chr  "No" "No" "No" "No" ...
##  $ BsmtFinType1  : chr  "Rec" "ALQ" "GLQ" "GLQ" ...
##  $ BsmtFinSF1    : num  468 923 791 602 263 0 935 0 637 804 ...
##  $ BsmtFinType2  : chr  "LwQ" "Unf" "Unf" "Unf" ...
##  $ BsmtFinSF2    : num  144 0 0 0 0 0 0 0 0 78 ...
##  $ BsmtUnfSF     : num  270 406 137 324 1017 ...
##  $ TotalBsmtSF   : num  882 1329 928 926 1280 ...
##  $ Heating       : chr  "GasA" "GasA" "GasA" "GasA" ...
##  $ HeatingQC     : chr  "TA" "TA" "Gd" "Ex" ...
##  $ CentralAir    : chr  "Y" "Y" "Y" "Y" ...
##  $ Electrical    : chr  "SBrkr" "SBrkr" "SBrkr" "SBrkr" ...
##  $ X1stFlrSF     : int  896 1329 928 926 1280 763 1187 789 1341 882 ...
##  $ X2ndFlrSF     : int  0 0 701 678 0 892 0 676 0 0 ...
##  $ LowQualFinSF  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ GrLivArea     : int  896 1329 1629 1604 1280 1655 1187 1465 1341 882 ...
##  $ BsmtFullBath  : num  0 0 0 0 0 0 1 0 1 1 ...
##  $ BsmtHalfBath  : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ FullBath      : int  1 1 2 2 2 2 2 2 1 1 ...
##  $ HalfBath      : int  0 1 1 1 0 1 0 1 1 0 ...
##  $ BedroomAbvGr  : int  2 3 3 3 2 3 3 3 2 2 ...
##  $ KitchenAbvGr  : int  1 1 1 1 1 1 1 1 1 1 ...
##  $ KitchenQual   : chr  "TA" "Gd" "TA" "Gd" ...
##  $ TotRmsAbvGrd  : int  5 6 6 7 5 7 6 7 5 4 ...
##  $ Functional    : chr  "Typ" "Typ" "Typ" "Typ" ...
##  $ Fireplaces    : int  0 0 1 1 0 1 0 1 1 0 ...
##  $ FireplaceQu   : chr  "0" "0" "TA" "Gd" ...
##  $ GarageType    : chr  "Attchd" "Attchd" "Attchd" "Attchd" ...
##  $ GarageYrBlt   : num  1961 1958 1997 1998 1992 ...
##  $ GarageFinish  : chr  "Unf" "Unf" "Fin" "Fin" ...
##  $ GarageCars    : num  1 1 2 2 2 2 2 2 2 2 ...
##  $ GarageArea    : num  730 312 482 470 506 440 420 393 506 525 ...
##  $ GarageQual    : chr  "TA" "TA" "TA" "TA" ...
##  $ GarageCond    : chr  "TA" "TA" "TA" "TA" ...
##  $ PavedDrive    : chr  "Y" "Y" "Y" "Y" ...
##  $ WoodDeckSF    : int  140 393 212 360 0 157 483 0 192 240 ...
##  $ OpenPorchSF   : int  0 36 34 36 82 84 21 75 0 0 ...
##  $ EnclosedPorch : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ X3SsnPorch    : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ ScreenPorch   : int  120 0 0 0 144 0 0 0 0 0 ...
##  $ PoolArea      : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ PoolQC        : chr  "0" "0" "0" "0" ...
##  $ Fence         : chr  "MnPrv" "0" "MnPrv" "0" ...
##  $ MiscFeature   : chr  "0" "Gar2" "0" "0" ...
##  $ MiscVal       : int  0 12500 0 0 0 0 500 0 0 0 ...
##  $ MoSold        : int  6 6 3 6 1 4 3 5 2 4 ...
##  $ YrSold        : int  2010 2010 2010 2010 2010 2010 2010 2010 2010 2010 ...
##  $ SaleType      : chr  "WD" "WD" "WD" "WD" ...
##  $ SaleCondition : chr  "Normal" "Normal" "Normal" "Normal" ...
##  $ SalePrice     : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ Foundation_q  : num  5 5 4 4 4 4 4 4 4 5 ...
##  $ BsmtFinType2_q: num  2 1 1 1 1 1 1 1 1 3 ...
##  $ HeatingQC_q   : num  3 3 4 5 5 4 5 4 4 3 ...
##  $ Electrical_q  : num  5 5 5 5 5 5 5 5 5 5 ...
##  $ KitchenQual_q : num  3 4 3 4 4 3 3 3 4 3 ...
##  $ GarageCond_q  : num  3 3 3 3 3 3 3 3 3 3 ...
##  $ Fence_q       : num  3 0 3 0 0 0 4 0 0 3 ...
kable(head(p2_test))
Id MSSubClass MSZoning LotFrontage LotArea Street Alley LotShape LandContour Utilities LotConfig LandSlope Neighborhood Condition1 Condition2 BldgType HouseStyle OverallQual OverallCond YearBuilt YearRemodAdd RoofStyle RoofMatl Exterior1st Exterior2nd MasVnrType MasVnrArea ExterQual ExterCond Foundation BsmtQual BsmtCond BsmtExposure BsmtFinType1 BsmtFinSF1 BsmtFinType2 BsmtFinSF2 BsmtUnfSF TotalBsmtSF Heating HeatingQC CentralAir Electrical X1stFlrSF X2ndFlrSF LowQualFinSF GrLivArea BsmtFullBath BsmtHalfBath FullBath HalfBath BedroomAbvGr KitchenAbvGr KitchenQual TotRmsAbvGrd Functional Fireplaces FireplaceQu GarageType GarageYrBlt GarageFinish GarageCars GarageArea GarageQual GarageCond PavedDrive WoodDeckSF OpenPorchSF EnclosedPorch X3SsnPorch ScreenPorch PoolArea PoolQC Fence MiscFeature MiscVal MoSold YrSold SaleType SaleCondition SalePrice Foundation_q BsmtFinType2_q HeatingQC_q Electrical_q KitchenQual_q GarageCond_q Fence_q
1461 20 RH 80 11622 Pave 0 Reg Lvl AllPub Inside Gtl NAmes Feedr Norm 1Fam 1Story 5 6 1961 1961 Gable CompShg VinylSd VinylSd None 0 TA TA CBlock TA TA No Rec 468 LwQ 144 270 882 GasA TA Y SBrkr 896 0 0 896 0 0 1 0 2 1 TA 5 Typ 0 0 Attchd 1961 Unf 1 730 TA TA Y 140 0 0 0 120 0 0 MnPrv 0 0 6 2010 WD Normal 0 5 2 3 5 3 3 3
1462 20 RL 81 14267 Pave 0 IR1 Lvl AllPub Corner Gtl NAmes Norm Norm 1Fam 1Story 6 6 1958 1958 Hip CompShg Wd Sdng Wd Sdng BrkFace 108 TA TA CBlock TA TA No ALQ 923 Unf 0 406 1329 GasA TA Y SBrkr 1329 0 0 1329 0 0 1 1 3 1 Gd 6 Typ 0 0 Attchd 1958 Unf 1 312 TA TA Y 393 36 0 0 0 0 0 0 Gar2 12500 6 2010 WD Normal 0 5 1 3 5 4 3 0
1463 60 RL 74 13830 Pave 0 IR1 Lvl AllPub Inside Gtl Gilbert Norm Norm 1Fam 2Story 5 5 1997 1998 Gable CompShg VinylSd VinylSd None 0 TA TA PConc Gd TA No GLQ 791 Unf 0 137 928 GasA Gd Y SBrkr 928 701 0 1629 0 0 2 1 3 1 TA 6 Typ 1 TA Attchd 1997 Fin 2 482 TA TA Y 212 34 0 0 0 0 0 MnPrv 0 0 3 2010 WD Normal 0 4 1 4 5 3 3 3
1464 60 RL 78 9978 Pave 0 IR1 Lvl AllPub Inside Gtl Gilbert Norm Norm 1Fam 2Story 6 6 1998 1998 Gable CompShg VinylSd VinylSd BrkFace 20 TA TA PConc TA TA No GLQ 602 Unf 0 324 926 GasA Ex Y SBrkr 926 678 0 1604 0 0 2 1 3 1 Gd 7 Typ 1 Gd Attchd 1998 Fin 2 470 TA TA Y 360 36 0 0 0 0 0 0 0 0 6 2010 WD Normal 0 4 1 5 5 4 3 0
1465 120 RL 43 5005 Pave 0 IR1 HLS AllPub Inside Gtl StoneBr Norm Norm TwnhsE 1Story 8 5 1992 1992 Gable CompShg HdBoard HdBoard None 0 Gd TA PConc Gd TA No ALQ 263 Unf 0 1017 1280 GasA Ex Y SBrkr 1280 0 0 1280 0 0 2 0 2 1 Gd 5 Typ 0 0 Attchd 1992 RFn 2 506 TA TA Y 0 82 0 0 144 0 0 0 0 0 1 2010 WD Normal 0 4 1 5 5 4 3 0
1466 60 RL 75 10000 Pave 0 IR1 Lvl AllPub Corner Gtl Gilbert Norm Norm 1Fam 2Story 6 5 1993 1994 Gable CompShg HdBoard HdBoard None 0 TA TA PConc Gd TA No Unf 0 Unf 0 763 763 GasA Gd Y SBrkr 763 892 0 1655 0 0 2 1 3 1 TA 7 Typ 1 TA Attchd 1993 Fin 2 440 TA TA Y 157 84 0 0 0 0 0 0 0 0 4 2010 WD Normal 0 4 1 4 5 3 3 0
# Predict prices for test data
#house_test <- read.csv('/Users/letiix3/Desktop/Data-605/Week-15/House_Price/test.csv')
p2_test_final <- p2_test %>%
  dplyr::select_if(is.numeric) %>%
  replace(is.na(.),0)

prediction <- predict(lm_final.lm, p2_test_final, type = "response")

head(prediction)
##        1        2        3        4        5        6 
## 108822.2 182340.8 185503.7 239686.1 171483.0 188038.8
# Preparing data frame for submission
kag_pred <- data.frame(Id = p2_test_final$Id, SalePrice = prediction)

head(kag_pred)
dim(kag_pred)
## [1] 1459    2
# commenting out to not create new file
#write.csv(kag_pred, file = "tns_submission_prediction.csv", row.names=FALSE)

#-![Kaggle Submission Confirmation!]("c:/r_images/kaggle_submission_data605.JPG")
Kaggle Confirmation

Kaggle Confirmation



References

https://mathworld.wolfram.com/ExponentialSumFormulas.html

https://pubs.wsb.wisc.edu/academics/analytics-using-r-2019/gamma-variables-optional.html

https://www.programmingr.com/examples/neat-tricks/sample-r-function/rexp/

https://bookdown.org/rdpeng/rprogdatascience/simulation.html

https://math.stackexchange.com/questions/2189317/mean-of-gamma-distribution

https://www.youtube.com/watch?v=cI-WFRqXbKM

https://www.pnw.edu/wp-content/uploads/2020/03/Lecture-Notes-7.pdf

https://www.tutorialspoint.com/set-values-in-categorical-column-to-numeric-values-in-r-data-frame

https://quantifyinghealth.com/vif-threshold/

---
title: "Data 605 Final"
subtitle: "CUNY SPS SPRING 2023"
date: "`r Sys.Date()`"
author: Tage Singh
output:
  rmdformats::readthedown:
    self_contained: false
    code_download: true
    toc_depth: 4
    df_print: paged
    code_folding: show
---


```{r setup, include=FALSE, warning=FALSE}

knitr::opts_chunk$set(echo = TRUE)

```

**Required Libraries**

```{r libs , message=FALSE, warning=FALSE}

library(data.table)
library(MASS)
library(Matrix)
library(matrixcalc)
library(dplyr)
library(ggplot2)
library(tidyverse)
library(purrr)
library(corrplot)
library(correlation)
library(knitr)
library(Hmisc)
library(forecast)
library(ggplot2)
library(ggthemes)
library(moments)
library(psych)
library(mctest)

```


---

# Problem 1

**Setting up the required Parameters :**

```{r parameters}

# Since we will use the same set of parameters for the 3 PDFs.

#set the seed - using this allows reproducibility of the sequence of random numbers

set.seed(68)


#We are required to choose a value of n > 3

n<- round(runif(1, 4, 100))


#We are required to choose a value of lambda between 2 and 10

lambda <- round(runif(1, 2, 10))


# The number of observations required is given by N :

N <- 10000

cat("We will use random generated values for 'n' and 'lambda' using the 'runif' function ","\n")

cat("The random value of n based on the requirement is : ","\n", (n))

cat("The random value of lambda based on the requirement is : ", "\n", (lambda))

cat("The required observations are : ", "\n", (N))

```

---

<br>

### Probability Density 1:  X~Gamma 


**Using R, generate a random variable $X$ that has 10,000 random Gamma Ɣ PDF values. A Gamma Ɣ PDF is completely describe by "n" (a size parameter) and lambda, λ (a shape parameter).  Choose any "n" greater  than (>) 3 and an expected value (λ) between 2 and 10 (you choose)**


```{r gamma}

#We will use the following function in R: rgamma(n, shape, rate = 1, scale = 1/rate)


cat("For n =",(n), ",  lambda = ",(lambda),"and ",(N),"observations : ")

xgamma <- rgamma(N, shape = n, rate = lambda)

cat("The first 10 values of the Gamma PDF are:", "\n", (head(xgamma,10)))

```


---

<br>

### Probability Density 2:  Y~Sum of Exponentials


Generate 10,000 observations from  the sum of $n$ exponential PDF with rate/shape parameter ($\lambda$). The $n$ and $\lambda$ must be the same as in the previous case. (e.g., $mysum$ $=$ $rexp$(10000,$\lambda$)+$rexp$(10000,$\lambda$))


```{r expsum}

# we will use the following function sum(rexp(n, lambda)), i.e. the sum of the rexp function

cat("For n =",(n), ",  lambda = ",(lambda),"and ",(N),"observations : ")

sumexp <- numeric(N)

for (i in 1:N) {
  sumexp[i] <- sum(rexp(n, lambda))
}

cat("The first 10 values of the Sum of Exponentials PDF are : ", "\n", (head(sumexp,10)))

```


---

<br>

### Probability Density 3:  Z~ Exponential


Generate 10,000 observations from  a single exponential pdf with rate/shape parameter ($\lambda$)

```{r exp}

expobs <- rexp(n = N, rate = lambda)

cat("For n =",(n), ",  lambda = ",(lambda),"and ",(N),"observations : ", "\n")

cat("The first 10 values of the Exponential PDF are : ", "\n", (head(expobs,10)))

```


---

<br>

# Problem 1a

**Calculate the empirical expected value (means) and variances of all three pdfs**

Note : The sample mean and variance are estimates of the population mean and variance, respectively, based on the  required sample of 10,000 observations.


```{r 1a}
# We will use the "mean" and "var" functions in R for this computation

cat("For n =",(n), "and lambda = ",(lambda),"and ",(N),"observations : ", "\n")

cat("The Empirical expected value (mean)  of the Gamma PDF is:", "\n", (mean(xgamma)))

cat("The Empirical variance of the Gamma PDF is:", "\n", (var(xgamma)))

cat("\n")

cat("\n")

cat("The Empirical expected value (mean)  of the Sum of Exponentials PDF is:", "\n", (mean(sumexp)))

cat("The Empirical variance of the Sum of Exponentials PDF is:", "\n", (var(sumexp)))

cat("\n")

cat("\n")

cat("The Empirical expected value (mean)  of the Exponential PDF is:", "\n", (mean(expobs)))

cat("The Empirical variance of the Sum of the Exponential PDF is:", "\n", (var(expobs)))

cat("\n")

```


---

<br>

# Problem 1b

**Using calculus, calculate the expected value and variance of the Gamma pdf (X).  Using the moment generating function for exponentials, calculate the expected value of the single exponential (Z) and the sum of exponentials (Y)**


### Probability Density 1:  X~Gamma 

 The Gamma Function is defined as : 
 
$\Gamma(\alpha) = \int_{0}^{\infty}y^{\alpha - 1}e^{-y} dy$$for$ $\alpha \gt 0$

The Expected Value of the Gamma Function is given as :

$E(X) = \int_{0}^{\infty} f(x) dx$

$\label{eq:gam-mean-s3}
\begin{split}
\mathrm{E}(X) &= \frac{a}{b} \int_{0}^{\infty} \mathrm{Gam}(x; a+1, b) \, \mathrm{d}x \\&= \frac{a}{b} \; .
\end{split}$

For our Computation, $a$ = $n$ = 93, and $b$ = $\lambda$ = 7 as was 
computed above--

$\implies$ $E(\Gamma)$ = $\frac{93}{7}$ = **13.28**

Similarly the Variance is given as $\frac{n}{\lambda^{2}}$ = $\frac{93}{7^{2}}$
= **1.89**


### Probability Density 3:  Z~ Exponential - Epected Value using MGF

the following proof is from TSingh - Assignment 9

The MGF of the Exponential Distribution is given by ;

${ g }_{ X }(t)=E({ e }^{ t }X)=\int _{ -\infty }^{ \infty }{ { e }^{ tx }{ f }_{ X }(x)dx. }$

$\Rightarrow$ $g(t)=\frac { λe^{ (t-λ)x } }{ t-λ } |_{ 0 }^{ ∞ }$

The First Moment

$\Rightarrow$ $g'(t)=\frac { λ }{ (λ-t)^{ 2 } }$

$\Rightarrow$ $g'(0)=\frac { λ }{ (λ-0)^{ 2 } }$

$\Rightarrow$ $g'(0) = \frac { λ }{ λ^{ 2 } } =\frac { 1 }{ λ }$ --- **The Expected Value - First Moment**


$\implies$ For our Computation The Expected Value - First Moment = 

$\frac { 1 }{ λ }$ = $\frac { 1 }{ 7 }$ 

= **0.143**


### Probability Density 2:  Y~Sum of Exponentials - Epected Value using MGF

--------------#####################------------------------


# 1c.  Probability

**For pdf Z (the exponential), calculate empirically probabilities a through c.  Then evaluate through calculus whether the memoryless property holds**


### a


For $P(Z>\lambda | Z>\frac{\lambda}{2})$ 

```{r 1ca}

Emp_prob_a <- 1-(pexp((mean(expobs)),lambda/2))

Emp_prob_a

```


### b


For $P(Z>2\lambda | Z>\lambda)$ 

```{r 1cb}

Emp_prob_b <- 1-(pexp((mean(expobs)),2*lambda))

Emp_prob_b

```

### c



For $P(Z>3\lambda | Z>\lambda)$ 

```{r 1cc}

Emp_prob_c <- 1-(pexp((mean(expobs)),3*lambda))

Emp_prob_c

```


# 1d
**Loosely investigate whether P(YZ) = P(Y) P(Z) by building a table with quartiles and evaluating the marginal and joint probabilities**

-----

# Problem 2

### Overview : https://www.kaggle.com/


Compete in the House Prices: Advanced Regression Techniques competition, provide r code for the following requirements :

Descriptive and Inferential Statistics. Provide univariate descriptive statistics and appropriate plots for the training data set.  Provide a scatterplot matrix for at least two of the independent variables and the dependent variable. Derive a correlation matrix for any three quantitative variables in the dataset.  Test the hypotheses that the correlations between each pairwise set of variables is 0 and provide an 80% confidence interval.  Discuss the meaning of your analysis.  Would you be worried about familywise error? Why or why not?


**Importing the data : **

```{r p2_data}

# Import provided datasets stored on Github
p2_train <- data.frame(read.csv('https://raw.githubusercontent.com/tagensingh/sps_data605_final_p_2/main/train.csv', header = T, sep = ","))
p2_test <- data.frame(read.csv('https://raw.githubusercontent.com/tagensingh/sps_data605_final_p_2/main/test.csv', header = T, sep = ","))
p2_test$SalePrice <- 0

```

###Descriptive Statistics

**Provide univariate descriptive statistics and appropriate plots for the training data set**

```{r descriptive}

# Summary of dataset : 

kable(head(p2_train))

# Dimension of Dataset : Rows X Columns

dim(p2_train)

# Structure of Columns

str(p2_train)

```


###Enhancing the Training Dataset

For our analysis we will convert some categorical columns to numerical values, using the values provided in the description file
These Operations will be done on both the Training and Test Datasets

```{r data_wrangle}

# Duplicating categorical columns and converting to numerical for additional analysis

# Adding Quantified Foundation Column

p2_train$Foundation_q = p2_train$Foundation
p2_train$Foundation_q <- c(Wood=1, Stone=2, Slab=3, PConc=4, CBlock=5, BrkTil=6)[p2_train$Foundation_q]


p2_test$Foundation_q = p2_test$Foundation
p2_test$Foundation_q <- c(Wood=1, Stone=2, Slab=3, PConc=4, CBlock=5, BrkTil=6)[p2_test$Foundation_q]


# Adding Quantified Basement Type Column

p2_train$BsmtFinType2_q = p2_train$BsmtFinType2
p2_train$BsmtFinType2_q <- c(Unf=1, LwQ=2, Rec=3, BLQ=4, ALQ=5, GLQ=6)[p2_train$BsmtFinType2_q]


p2_test$BsmtFinType2_q = p2_test$BsmtFinType2
p2_test$BsmtFinType2_q <- c(Unf=1, LwQ=2, Rec=3, BLQ=4, ALQ=5, GLQ=6)[p2_test$BsmtFinType2_q]


# Adding Quantified Heating Quality Column

p2_train$HeatingQC_q = p2_train$HeatingQC
p2_train$HeatingQC_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_train$HeatingQC_q]	

p2_test$HeatingQC_q = p2_test$HeatingQC
p2_test$HeatingQC_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_test$HeatingQC_q]


# Adding Quantified Electrical Quality Column

p2_train$Electrical_q = p2_train$Electrical
p2_train$Electrical_q <- c(Mix=1, FuseP=2, FuseF=3, FuseA=4, SBrkr=5)[p2_train$Electrical_q]

p2_test$Electrical_q = p2_test$Electrical
p2_test$Electrical_q <- c(Mix=1, FuseP=2, FuseF=3, FuseA=4, SBrkr=5)[p2_test$Electrical_q]


# Adding Quantified KitChen Quality Column

p2_train$KitchenQual_q = p2_train$KitchenQual
p2_train$KitchenQual_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_train$KitchenQual_q]

p2_test$KitchenQual_q = p2_test$KitchenQual
p2_test$KitchenQual_q <- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_test$KitchenQual_q]


# Adding Quantified Garage Condition Column

p2_train$GarageCond_q = p2_train$GarageCond
p2_train$GarageCond_q<- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_train$GarageCond_q]

p2_test$GarageCond_q = p2_test$GarageCond
p2_test$GarageCond_q<- c(Po=1, Fa=2, TA=3, Gd=4, Ex=5)[p2_test$GarageCond_q]


# Adding Quantified Fence Condition Column

p2_train$Fence_q = p2_train$Fence
p2_train$Fence_q <- c(MnWw=1, GdWo=2, MnPrv=3, GdPrv=4)[p2_train$Fence_q]

p2_test$Fence_q = p2_test$Fence
p2_test$Fence_q <- c(MnWw=1, GdWo=2, MnPrv=3, GdPrv=4)[p2_test$Fence_q]


# Some Cleanup to account for reserved "NA"

p2_train <- replace(p2_train,is.na(p2_train),0)

p2_test <- replace(p2_test,is.na(p2_test),0)


# Printing the enhanced data frame with new quantitative columns

p2_train %>% select(order(colnames(p2_train)))


str(p2_train)


kable(head(p2_train))

```


### Investigative Scatter Plots



```{r}

p2_train$OverallCond_factor <- as.factor(as.character(p2_train$OverallCond))
ggplot(p2_train, aes(x=OverallCond, y=SalePrice, fill=OverallCond_factor)) + geom_boxplot()

p2_train$OverallCond_factor<-NULL
```

```{r}
ggplot(p2_train, aes(x=Neighborhood, y=SalePrice, fill=Neighborhood)) + geom_boxplot()+ coord_flip()

```

A graphical view of the Sales Price Spread vs Year Remodeled
Note that We used the year Remodeled vs Year Built since for the homes that were not Remodeled, the year built was used.

```{r sp_spread}


ggplot(p2_train, aes(x = YearRemodAdd, y = SalePrice)) +
  geom_point()+
  geom_smooth(method=lm) +
  scale_y_continuous(labels = scales::comma)

```

### Correlation Matricies


**Derive a correlation matrix for any three quantitative variables in the dataset**

Note: as a step further we will compute the Correlation Matrix for a range of quantitative variables below

```{r corr}

corr_data<-dplyr::select(p2_train,SalePrice,LotArea,BsmtFinSF2,GarageArea,YearRemodAdd,OverallCond,TotalBsmtSF,GrLivArea,HeatingQC_q,Electrical_q,KitchenQual_q,Fence_q,GarageCond_q)

corr_matrix<-round(cor(corr_data),4)

#Correlation Matrix with correlation matrix coefficients
corrplot(corr_matrix, method = 'number') # colorful number

# Another Visual of the Correlation Matrix

corrplot(corr_matrix, order = 'hclust', addrect = 2)

```

<br>

### Correlation Hypothesis Testing

We are required to compute 3 pairs, we will compute an additional 3 pairs to solidify the concept


#### Sales Price Vs Year Remodeled

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.481 and 0.531. The sample estimate is +0.51



```{r ch_sp_yr_rm}

cor.test(corr_data$SalePrice,corr_data$YearRemodAdd, conf.level = 0.8)

```

#### Sales Price Vs Lot Area

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.232 and 0.294 The sample estimate is +0.26


```{r ch_sp_lot_area}

cor.test(corr_data$SalePrice,corr_data$LotArea, conf.level = 0.8)

```

#### Sales Price Vs Overall Condition

This pair of variable computes a low P-Value (0.002) indication a likely non zero(0) correlation and 80% confidence that the correlation is between -0.111 and -0.044 The sample estimate is -0.07, This indicates a zero to slight inverse relationship between Sales Price and Overall Condition.


```{r ch_sp_condit}

cor.test(corr_data$SalePrice,corr_data$OverallCond, conf.level = 0.8)

```

#### Sales Price Vs Quality of Heating System

The Heating Quality variable is derived from converting a categorical field to a numeric

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.399 and 0.454 The sample estimate is +0.427


```{r ch_sp_heat}

cor.test(corr_data$SalePrice,corr_data$HeatingQC_q, conf.level = 0.8)

```

#### Sales Price Vs Kitchen Quality

The Kitchen Quality variable is derived from converting a categorical field to a numeric

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between 0.640 and 0.678 The sample estimate is +0.659


```{r ch_sp_kitc}

cor.test(corr_data$SalePrice,corr_data$KitchenQual_q, conf.level = 0.8)

```

#### Sales Price Vs Fence Condition

The Fence Quality variable is derived from converting a categorical field to a numeric

This pair of variable computes a low P-Value indication a likely non zero(0) correlation and 80% confidence that the correlation is between -0.179 and -0.113 The sample estimate is -0.146, This indicates an inverse relationship between Sales Price and Overall Condition


```{r ch_sp_fence}

cor.test(corr_data$SalePrice,corr_data$Fence_q, conf.level = 0.8)

```

**Pairwise Correlation Discussion**

The six correlation analysis pairs of variables show that correlation exist between the Sales Price (Dependent Variable) and the Independent Variables examined. There are some strong correlation in the 80% confidence interval except for the "Fence Condition" and "Overall Condition" Variables which indicated a zero to slight inverse relationship to the Sales Price. This quantitative evidence is not worrying with respect to familywise errors.

---


## Linear Algebra and Correlation


**Inverting the Correlation Matrix to Create Precision Matrix**

```{r invert_corr}
##    The Current Correlation Matrix

corr_matrix

##    Inverting the correlation Matrix to Create the precision Matrix

Invert_matrix<-round(solve(corr_matrix),4)


##   Creating the Precision Matrix 

###   corr X invert

precision_matrix_1 <- round(corr_matrix %*% Invert_matrix,4)

precision_matrix_1


###   invert X corr

precision_matrix_2 <- round(Invert_matrix %*% corr_matrix,4)

precision_matrix_2

```


```{r decomp_pm1}

###   The Decomposition of precision_matrix_1 is :

decomp_pm1 <- lu.decomposition(precision_matrix_1)

decomp_pm1

```


```{r decomp_pm2}


###   The Decomposition of precision_matrix_1 is :

decomp_pm2 <- lu.decomposition(precision_matrix_2)

decomp_pm2

```

```{r mat_comp}

## Comparing the 2 Matrices, we see that they are equal ( when rounded to 3 decimal places)

round(precision_matrix_1 ,3)== round(precision_matrix_2,3)

```


---

## Calculus-Based Probability & Statistics

**Many times, it makes sense to fit a closed form distribution to data. Select a variable in the Kaggle.com training dataset that is skewed to the right, shift it so that the minimum value is absolutely above zero if necessary**

```{r skew_check}

head(corr_data)

skew(corr_data, na.rm = TRUE)

## We see that "LotArea" field is the most RIGHT skewed with a value of :

round(skew(corr_data$LotArea, na.rm = TRUE),3)

## A Histogram of the field :


ggplot(corr_data, aes(x=LotArea)) + geom_histogram(color="blue", fill="white", binwidth = 1000)+labs(title="Lot Area plot - Skewness = 12.2 - Min Value = 1300",x="Lot Size", y = "Count")

```


### The Fitting

**Then load the MASS package and run fitdistr to fit an exponential probability density function**


```{r fitting}

la_fit <- corr_data$LotArea

summary(la_fit) ### Note that the minimum value is > 0


###  Determining the fit

fit <- fitdistr(la_fit, "exponential")

fit


```


**Find the optimal value of λ for this distribution, and then take 1000 samples from this exponential distribution using this value**


```{r sampling}

# Computing Lambda
lambda_fit <- fit$estimate

lambda_fit

### Generating new distribution and Histogram 

new_dist <- rexp(1000, lambda_fit)

summary(new_dist)

hist(new_dist,breaks = 100)

```


**Plot histogram and compare it with original histogram**

Note: As shown below, using the Lambda from the "LotArea"  variable and applying 
it a new Exponential distribution yields a similar histogram, the differences in 
distribution values has some effect on the resulting histogram but not significant.

```{r comp_plots}

fit_df <- data.frame(length = la_fit)
new_dist_df <- data.frame(length = new_dist)

fit_df$from <- 'Fit'
new_dist_df$from <- 'New Dist'

both_df <- rbind(fit_df,new_dist_df)

ggplot(both_df, aes(length, fill = from)) + geom_density(alpha = 0.5)

```


---

**Using the exponential pdf, find the 5th and 95th percentiles using the cumulative distribution function (CDF). Also generate a 95% confidence interval from the empirical data, assuming normality. Finally, provide the empirical 5th percentile and 95th percentile of the data. Discuss.**

The Exponential PDF is given as $f(x;\lambda) = \lambda e^{-\lambda x}$ for $x \geq 0$

The CDF is given as  $f(x;\lambda)=1−e^{-\lambda x}$

$\lambda$ is given as : 9.50857

To find the $5^{th}$ percentile we solve for x in :

$0.05 = 1 - e^{\lambda x}$

$\implies$ $0.05 = 1 - e^{-\lambda x}$

$\implies$ $-ln(0.95) = \lambda x$

$\implies$ $ x = \frac{-ln(0.95)}{\lambda}$


To find the $95^{th}$ percentile we solve for x in :

$0.95 = 1 - e^{\lambda x}$

$\implies$ $0.95 = 1 - e^{-\lambda x}$

$\implies$ $-ln(0.05) = \lambda x$

$\implies$ $ x = \frac{-ln(0.05)}{\lambda}$


```{r percentiles}
percent_5th <- round((-log(0.95)/lambda_fit),4)

cat("The 5th Percentile is given as : ", "\n", (percent_5th))


percent_95th <- round((-log(0.05)/lambda_fit),4)

cat("The 95th Percentile is given as : ", "\n", (percent_95th))


##    To Compute 95% confidence interval from the empirical data

mean_la_fit <-mean(la_fit)

p2_norm<-rnorm(length(la_fit),mean(la_fit),sd(la_fit))

cat("The 95th confidence interval from the data is given as : ", "\n",(quantile(p2_norm, probs=c(0.05, 0.95))))

#  The Histogram of the distribution is :

hist(p2_norm)

## The empirical 5th percentile and 95th percentile of the data is given as :

quantile(la_fit, c(0.05, 0.95))

```


---

# Modeling

**Build some type of multiple regression  model and submit your model to the competition board.  Provide your complete model summary and results with analysis**


```{r data_check}
## Overview of datasets including the addition converted categorical columns.

# Printing the enhanced data frame with new quantitative columns

# ------ Training Dataset

p2_train %>% select(order(colnames(p2_train)))

str(p2_train)

kable(head(p2_train))


# ------ Test Dataset

p2_test %>% select(order(colnames(p2_test)))

str(p2_test)

kable(head(p2_test))

```


Preparing the Training dataset by removing the non-numerical columns

```{r data_prep}

# selection columns that are numeric only
p2_train_num <- p2_train %>% 
  dplyr::select_if(is.numeric)

# Dropping the "id" and "SalePrice" fields since it is not needed for the predictor model Variables

p2_train_vars <- subset(p2_train_num, select = -c(Id,SalePrice))

# Check for missing values in data 

colSums(is.na(p2_train_vars))

## Reviewing the structure of the enhanced dataset

str(p2_train_vars)

dim(p2_train_vars)

kable(head(p2_train_vars))

```
Since our initial data preparation yielded 43 numerical variables that are 
eligible to be included in the linear model, we will use additional tools
to narrow the selection to variables that will yield "best fit" results.
The two computations to be employed are :

Test for Multicollinearity 

Test for Correlation to the predicted variable **- Sales Price -**

```{r variable_selection}

##  the summary of the dataset is : 

summary(p2_train_vars)

## Checking for Null Values 

p2_train_vars[!complete.cases(p2_train_vars),]


## Predictor Variables included in initial regression : 

sort(colnames(p2_train_vars))



```

### Regression Model Fitting

**We will perform Regression Modeling and manipulate predictor variables to compute the optimal outcome**

**Regression Modeling V1 and V2**

Note that after computing Linear Model v2, we have Multiple - $R^{2} = 0.823$
and $R^{2} = 0.8183$

```{r regression_modeling_v1_v2}

##### Note that object "p2_train_vars" holds all predictor variables- columns

p2_train_regr_v1 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars)), collapse = "+"),
        sep = ""
    ))

## The Resultant variable list :

p2_train_regr_v1

#--------------- Linear Model Version 1 --------------

lm_1 <- lm((p2_train_regr_v1),data = p2_train)

summary(lm_1)

###############################################

# Removing "TotalBsmtSF" and "X2ndFlrSF" from  "p2_train_vars"

p2_train_vars_2 <- subset(p2_train_vars, select = -c(TotalBsmtSF,X2ndFlrSF))

p2_train_regr_v2 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_2)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v2

#--------------- Linear Model Version 2 --------------

lm_2.lm <- lm((p2_train_regr_v2),data = p2_train)

summary(lm_2.lm)

###############################################


```


---


**Testing for and Freeing From  Multicollinearity among Variables**

Multicollinearity  occurs when two or more predictor variables are highly correlated to each other, such that they do not provide unique or independent information in the regression model.

If the degree of correlation is high enough between variables, it can cause problems when fitting and interpreting the regression model.

To test this model for Multicollinearity we will employ the "imcdiag" function from the "mctest" library and examine the 
Variance Inflation Factor (VIF) score.

Note : Scores over 5 are moderately multicollinear. Scores over 10 are very problematic

using the VIF measure we see that most of the predictor variables posses low VIF scores indicating that they are not very correlated, but the following variables are moderately to problematic :


GarageCond_q - VIF score is 11.3 ---- Problematic ---- Will be removed from the model

GarageYrBlt - VIF score is 11.60 ---- Problematic ---- Will be removed from the model

GrLivArea- VIF score is 8.48 ---- moderately multicollinear ---- 



```{r collinear_test_1}

imcdiag(lm_2.lm)

```

---

**Regression Modeling V3**


```{r regression_modeling_v3}

# We will remove the "GarageCond_q" and "GarageYrBlt" variables and recompute the Linear Model

p2_train_vars_3 <- subset(p2_train_vars_2, select = -c(GarageCond_q,GarageYrBlt))

p2_train_regr_v3 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_3)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v3

#--------------- Linear Model Version 3 --------------

lm_3.lm <- lm((p2_train_regr_v3),data = p2_train)

summary(lm_3.lm)

###############################################

imcdiag(lm_3.lm)


```
**Version 3 Discussion :**

Note that after computing Linear Model V3, we have Multiple - $R^{2} = 0.8198$
and $R^{2} = 0.8149$ - Not a significant difference from V2 modeling

Also - The Multicollinearity test shows the VIF scores for the following variables to be > 8

"GrLivArea" 

These will be removed in V4 :


---

**Regression Modeling V4**


```{r regression_modeling_v4}

# We will remove the "GrLivArea" variables : 

p2_train_vars_4 <- subset(p2_train_vars_3, select = -c(GrLivArea))

p2_train_regr_v4 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_4)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v4

#--------------- Linear Model Version 4 --------------

lm_4.lm <- lm((p2_train_regr_v4),data = p2_train)

summary(lm_4.lm)



imcdiag(lm_4.lm)


```
**Version 4 Discussion :**

Note that after computing Linear Model V4, we have Multiple - $R^{2} = 0.8099$
and $R^{2} = 0.8049$ - 

Note - As a final tuning to the model, we will remove the variables with 
VIF Scores > 3 in V5 of the Model. - Taking a more Conservative Approach as 
suggested by some researchers - https://quantifyinghealth.com/vif-threshold/



---

**Regression Modeling V5**


```{r regression_modeling_v5}

# We will remove the "BsmtFinSF1", "BsmtFinSF2, "BsmtUnfSF", "GarageArea", "GarageCars", "OverallQual", "TotRmsAbvGrd", "X1stFlrSF", "YearBuilt" variables : 

p2_train_vars_5 <- subset(p2_train_vars_4, select = -c(BsmtFinSF1,BsmtFinSF2,BsmtUnfSF,GarageArea,GarageCars,OverallQual,TotRmsAbvGrd,X1stFlrSF,YearBuilt))

p2_train_regr_v5 <- as.formula(paste("SalePrice", "~",
        paste(sort(colnames(p2_train_vars_5)), collapse = "+"),
        sep = ""
    ))

p2_train_regr_v5

#--------------- Linear Model Version 5 --------------

lm_5.lm <- lm((p2_train_regr_v5),data = p2_train)

summary(lm_5.lm)



imcdiag(lm_5.lm)


```
**Version 5 Discussion :**

Note that after computing Linear Model V5, we have Multiple - $R^{2} = 0.7183$
and $R^{2} = 0.7126$ - A Decline - Not in the Expected Direction

---



## Final Model Adjustments

**We will test our model with a low score stepAIC model**

stepAIC is one of the most commonly used search method for feature selection. We try to keep on minimizing the stepAIC value to come up with the final set of features. “stepAIC” does not necessarily mean to improve the model performance, however, it is used to simplify the model without impacting much on the performance.

We will use the stepAIC procedure to determine the final model components !

This is based on a model prediction with the lowest AIC score :

the scores for our model range from AIC=31154.79 to AIC=31141.34

We will be using the model with score : AIC=31141.34

```{r stepaic}

#### - StepAIc

stepAIC(lm_5.lm, direction="both")

```


---

### Final Model

This model is the result of 5 iterations of the original model followed by 
a stepAIC computation producing the Final Model as follows :

```{r final_model}

## The following model computed the lowest score of AIC=31141.34

lm_final.lm <- lm((SalePrice ~ BsmtFullBath + Fence_q + Fireplaces + Foundation_q + FullBath + HalfBath + HeatingQC_q + KitchenAbvGr + KitchenQual_q + LotArea + LotFrontage + LowQualFinSF + MasVnrArea + MSSubClass + OpenPorchSF + OverallCond + ScreenPorch + WoodDeckSF + YearRemodAdd + YrSold), data = p2_train)

summary(lm_final.lm)

```


**Residuals Discussion**

The histogram of the residuals shows an almost perfect normal distribution with mean approximately = 0

In the QQ lot, data that aligns closely to the red line indicates a normal distribution. If the points skew drastically from the line, you could consider adjusting your model by adding or removing other variables in the regression model, this model is the result of that model adjustment.


```{r residuals_1}

hist(lm_final.lm$residuals, prob = TRUE)
abline(v = mean(lm_final.lm$residuals),                       # Add line for mean
       col = "red",
       lwd = 3)
lines(density(lm_final.lm$residuals),col = "blue")

```



```{r residuals_2}

qqnorm(lm_final.lm$residuals)
qqline(lm_final.lm$residuals, col = "red")


```

The fitted and residual values seem to have a linear relationship, there is some evidence of heteroskedastic behavior



```{r residuals_3}

plot(lm_final.lm$fitted.values, lm_final.lm$residuals, 
     xlab="Fitted Values", ylab="Residuals",
     main="Residuals Plot",col = "blue")
abline(h=0)

```

---

## Predicting the Test Data

```{r predictions}


p2_test %>% select(order(colnames(p2_test)))

str(p2_test)

kable(head(p2_test))



# Predict prices for test data
#house_test <- read.csv('/Users/letiix3/Desktop/Data-605/Week-15/House_Price/test.csv')
p2_test_final <- p2_test %>%
  dplyr::select_if(is.numeric) %>%
  replace(is.na(.),0)

prediction <- predict(lm_final.lm, p2_test_final, type = "response")

head(prediction)


# Preparing data frame for submission
kag_pred <- data.frame(Id = p2_test_final$Id, SalePrice = prediction)

head(kag_pred)

dim(kag_pred)

# commenting out to not create new file
#write.csv(kag_pred, file = "tns_submission_prediction.csv", row.names=FALSE)

#-![Kaggle Submission Confirmation!]("c:/r_images/kaggle_submission_data605.JPG")

```

```{r figurename, echo=FALSE, fig.cap="Kaggle Confirmation", out.width = '150%'}
knitr::include_graphics("kaggle_submission_data605.JPG")
```


---


<br>

**References**

https://mathworld.wolfram.com/ExponentialSumFormulas.html


https://pubs.wsb.wisc.edu/academics/analytics-using-r-2019/gamma-variables-optional.html


https://www.programmingr.com/examples/neat-tricks/sample-r-function/rexp/


https://bookdown.org/rdpeng/rprogdatascience/simulation.html


https://math.stackexchange.com/questions/2189317/mean-of-gamma-distribution


https://www.youtube.com/watch?v=cI-WFRqXbKM


https://www.pnw.edu/wp-content/uploads/2020/03/Lecture-Notes-7.pdf


https://www.tutorialspoint.com/set-values-in-categorical-column-to-numeric-values-in-r-data-frame

https://quantifyinghealth.com/vif-threshold/
